A (new) prime theorem.
 

"A Prime Number Theorem" was published in 1986
(ISBN 87 7245 129 7, Rhodos Publishers Copenhagen, DK).

"Possible primes" were defined as [(6*m)+1], m being an integer
from - infinity to + infinity.
Negative possible primes (-5,-11,-17,-23.....) have modules V, II or VIII.
Positive possible primes (1,7,13,19,25,....) have modules I, VII or IV.

The integers 2 and 3 cannot be defined as possible primes (6*m +1)
and should not be considered as primes.

The integer 1 is a square (6*0 +1)*(6*0 +1), just as 25 is equal to
[(6*(-1) +1)] * [(6*(-1) +1)]
and 49 is equal to [(6*(+1) +1)] * [(6*(+1) +1)].

Products of possible primes remain possible primes
36 * (n*m) + 6* (n+m) +1, n being an integer from - infinity to + infinity.

All Mersenne primes are positive possible primes and will be defined
in a later thread.

Troels Munkneer

]

[quote=troels munkner;88242]The integers 2 and 3 cannot be defined as possible primes (6*m +1)
and should not be considered as primes.[/quote]Are you saying that 2 and 3 are not prime and are therefore composite? Do you happen to know of an integer that divides either of them besides 1? :rolleyes:[quote=troels munkner;88242][(6*(+1) +1)] * [(6*(+1) +1)].[/quote]The syntax of this is incorrect. You cannot say (+1), it is just plain incorrect. What do you mean by this? :ermm:

[QUOTE="troels munkner"]All Mersenne primes are positive possible primes[/QUOTE]For all n=odd, n>=1 the following will hold ((2^n)-1)=6m+1. But what does that prove? What are you trying to say?

troels munkner doesn't follow standard definitions and conventions, and his work is not supported by others.
I suggest moving this thread to Miscellaneous Math Threads. If I were a moderator, I would ask him to only post about his "possible primes" there. Maybe merge this thread with some of his similar unsupported stuff [URL="http://mersenneforum.org/showthread.php?t=5937"]here[/URL] or [URL="http://mersenneforum.org/showthread.php?t=6090"]here[/URL]. And delete his duplicate post in Information & Answers.

[QUOTE=troels munkner;88242]"A Prime Number Theorem" was published in 1986
(ISBN 87 7245 129 7, Rhodos Publishers Copenhagen, DK).[/quote]
If that is true, and APNT is basically the same bogus crap you keep posting to this board every few months, it seems their editorial standards are "colossally" low.

[quote]The integers 2 and 3 cannot be defined as possible primes (6*m +1)
and should not be considered as primes.[/quote]
You have a most curious definition of "possible prime." Note that I do not intend "curious" to imply in any way that your "definition" is interesting - rather, the kinds of descriptive terms that do come to mind include "idiotic," "clueless", and "wasteful of other people's time." Might I suggest that you either keep your inane musings to yourself, or take them elsewhere? I normally would refrain from using such harsh language, but this is not the first time you've posted this garbage here.

[quote]The integer 1 is a square (6*0 +1)*(6*0 +1), just as 25 is equal to
[(6*(-1) +1)] * [(6*(-1) +1)]
and 49 is equal to [(6*(+1) +1)] * [(6*(+1) +1)].[/quote]
And this tells us what, exactly? That certain perfect squares are ... perfect squares?

[quote]Products of possible primes remain possible primes
36 * (n*m) + 6* (n+m) +1, n being an integer from - infinity to + infinity.[/quote]
So the primes 2 and 3 are in fact not prime, at the same time that any product of your designated 6k+1 possible primes is not clearly composite? You sir, are a moron.

[quote]All Mersenne primes are positive possible primes and will be defined
in a later thread.[/quote]
I tremble at the thought of the further "enlightenment" you speak of.

[QUOTE=Jens K Andersen;88301]troels munkner doesn't follow standard definitions and conventions, and his work is not supported by others.
I suggest moving this thread to Miscellaneous Math Threads. If I were a moderator, I would ask him to only post about his "possible primes" there. Maybe merge this thread with some of his similar unsupported stuff [URL="http://mersenneforum.org/showthread.php?t=5937"]here[/URL] or [URL="http://mersenneforum.org/showthread.php?t=6090"]here[/URL]. And delete his duplicate post in Information & Answers.[/QUOTE]

a polite dialogue will be appreciated
All integers from - infinity to + infinity can be subdivided into three groups.
A. Even integers which will be products of 2 and an other integer.
B. Odd integers divisible by 3 which will be products of 3 and an other
odd integer.
C. Odd integers which are not divisible by 3.
Their general form is (6*m +1), m being an integer from - infinity
to + infinity.
These integers can be grouped as "possible primes" and comprise
real primes and possible prime products.
All possible primes are "located" along a straight line wuith an individual
difference of 6, i.e. (6*m +1) --- (-35), (-29), (-23), (-17), (-11), (-5),
1,7,13,19,25,31,37 --- (6*m +1)
Possible primes constitute exactly one third of allo integers.
By modulation (modulo 9) possible primes have modules II,V,VIII,or
I,IV,VII.

Y.s.

Troels Munkner

[QUOTE=troels munkner;88444]a polite dialogue will be appreciated
All integers from - infinity to + infinity can be subdivided into three groups.
A. Even integers which will be products of 2 and an other integer.
B. Odd integers divisible by 3 which will be products of 3 and an other
odd integer.
C. Odd integers which are not divisible by 3.
Their general form is (6*m +1), m being an integer from - infinity
to + infinity.
These integers can be grouped as "possible primes" and comprise
real primes and possible prime products.
All possible primes are "located" along a straight line wuith an individual
difference of 6, i.e. (6*m +1) --- (-35), (-29), (-23), (-17), (-11), (-5),
1,7,13,19,25,31,37 --- (6*m +1)
Possible primes constitute exactly one third of allo integers.
By modulation (modulo 9) possible primes have modules II,V,VIII,or
I,IV,VII.

Y.s.

Troels Munkner[/QUOTE]This statement is both correct and trivial. It also defines the term "possible prime" to mean integers of the form [tex]6k\pm1[/tex]. No-one but you uses the term "possible prime" but that's ok.

Where you go seriously off the rails is your claim that neither 2 nor 3 is a prime number. By making this statement you are not using the word "prime" in the same sense as it is used by essentially all mathematicians.

Like Humpty Dumpty you are at liberty to use whatever words you wish with whatever meaning you choose to assign to them The downside of that freedom is that if you use words with a meaning different from that understood by everyone else, you can guarantee that no-one will understand you.

If your objective is to annoy others and/or make yourself look stupid --- fair enough, though we have the freedom to ridicule you and/or express our annoyance. On the other hand, if you wish to communicate your ideas it is a very good idea to use a common language and that includes using words which have mutually agreed meaning.


Paul

[QUOTE=troels munkner;88444]a polite dialogue will be appreciated
All integers from - infinity to + infinity can be subdivided into three groups.
A. Even integers which will be products of 2 and an other integer.
B. Odd integers divisible by 3 which will be products of 3 and an other
odd integer.
C. Odd integers which are not divisible by 3.
Their general form is (6*m +1), m being an integer from - infinity
to + infinity.
These integers can be grouped as "possible primes" and comprise
real primes and possible prime products.
All possible primes are "located" along a straight line wuith an individual
difference of 6, i.e. (6*m +1) --- (-35), (-29), (-23), (-17), (-11), (-5),
1,7,13,19,25,31,37 --- (6*m +1)
Possible primes constitute exactly one third of allo integers.
[/QUOTE]

If I understand well 5 is not an integer since it is not Even, not divisible by 3 and not of the form 6 * m + 1.

So all numbers of the form 6 * m + 5 (where m is an integer) are not integers.

I can deduce from that, that 31 for instance can not be an integer, and thus not be a "possible prime", since it would require m = 5 to get 6 * 5 + 1 = 31.

[QUOTE=Jacob Visser;88449]If I understand well 5 is not an integer since it is not Even, not divisible by 3 and not of the form 6 * m + 1.

So all numbers of the form 6 * m + 5 (where m is an integer) are not integers.

I can deduce from that, that 31 for instance can not be an integer, and thus not be a "possible prime", since it would require m = 5 to get 6 * 5 + 1 = 31.[/QUOTE]

Note that the integers that are 1 mod 6 form an algebra in which unique
multiplication FAILS. Take the number 3025. It is in the set. So are
25 (a prime), 121 (a prime) and 55 (a prime).

But 3025 = 25*121 (product of two primes)
= 55*55 (square of a different prime!)

So we have a number that is the square of a prime also equal to the
product of two different primes!

[QUOTE=xilman;88448]This statement is both correct and trivial. It also defines the term "possible prime" to mean integers of the form [tex]6k\pm1[/tex]. No-one but you uses the term "possible prime" but that's ok.

Where you go seriously off the rails is your claim that neither 2 nor 3 is a prime number. By making this statement you are not using the word "prime" in the same sense as it is used by essentially all mathematicians.
Paul[/QUOTE]
:rolleyes:
The only reason I can presume to explain this is that 1 is not considered a prime. It is its own square and this property is unique.

Since 2 is considered the only even prime it 'may' also be dropped out of the 'real' prime sequence.

Now Goldbach's conjecture is that every even number greater the two (2=1+1) is the sum of 2 prime numbers. So two is not, by definition above of 1, not being a prime and Goldbach makes 2 an exception to his rule..

Is that what you mean Troels?

But why do you consider 3 as not a prime number? Have you a logical reason?

Mally :coffee:

[QUOTE=R.D. Silverman;88453]Note that the integers that are 1 mod 6 form an algebra in which unique
multiplication FAILS. Take the number 3025. It is in the set. So are
25 (a prime), 121 (a prime) and 55 (a prime).[/QUOTE]
See, if the original poster used language more like that (or had anything meaningful to say), we might take him more seriously...

standard definitions
 

[QUOTE=Jens K Andersen;88301]troels munkner doesn't follow standard definitions and conventions, and his work is not supported by others.
I suggest moving this thread to Miscellaneous Math Threads. If I were a moderator, I would ask him to only post about his "possible primes" there. Maybe merge this thread with some of his similar unsupported stuff [URL="http://mersenneforum.org/showthread.php?t=5937"]here[/URL] or [URL="http://mersenneforum.org/showthread.php?t=6090"]here[/URL]. And delete his duplicate post in Information & Answers.[/QUOTE]
:smile:
Jens we must remember that new concepts originate from older ones and therefore new terms of description require rigid definitions. Once these are defined they have to be incorporated for the new concept.
Maybe Troels finds it difficult to describe his concepts and we must have patience and give him the benefit of the doubt.

By condemning his efforts, it will not get us anywhere.

Also we should be willing to imbibe strange concepts which is the true spirit of scientific research.

I had not seen a contour integral till I had to refer to books on topology.
Maybe it is old hat for some but being new to it I had to accept it.

You may be aware of the great battle between Newton and Leibniz on symbols. They both were allowed, and not mercilessly stamped out, as each meant the same . Its only later that Leibnitz' symbols were more practical to work with and Newton's were relegated to dynamics .

The aim of education is to draw out a concept in such a way that the person questioned himself gives the answer.

My appeal is to have more patience and tolerance.

To me the person who straight away condemns a concept is too lazy to think for himself or change his inbuilt way of thinking.

Mally :coffee:

[QUOTE=mfgoode;88454]:rolleyes:
The only reason I can presume to explain this is that 1 is not considered a prime. It is its own square and this property is unique.

Since 2 is considered the only even prime it 'may' also be dropped out of the 'real' prime sequence.

Now Goldbach's conjecture is that every even number greater the two (2=1+1) is the sum of 2 prime numbers. So two is not, by definition above of 1, not being a prime and Goldbach makes 2 an exception to his rule..

Is that what you mean Troels?

But why do you consider 3 as not a prime number? Have you a logical reason?

Mally :coffee:[/QUOTE]

Once one starts to eliminate numbers from the prime list nothing remains from the Goldbach conjecture : if one can not use one, nor two, nor three, it implies that four, six and eight are not the sums of two primes.

Obviously the arguments to question the "troelsian" arithmetic do not need to be very elaborate.

The only good thing about Troels posts is that it permits me to participate on a discussion about mathematics ;-) I do not believe that new insights wil be achieved just by defining a different language that does not bring anything new but confusion.

[QUOTE=R.D. Silverman;88453]Note that the integers that are 1 mod 6 form an algebra in which unique multiplication FAILS. [/QUOTE]Minor typo? I think you meant unique [b]factorization[/b].


Paul

[QUOTE=mfgoode;88457]Maybe Troels finds it difficult to describe his concepts and we must have patience and give him the benefit of the doubt.[/QUOTE]

Sorry, but this is not exactly a new field of mathematics we're talking about here - it's been perfectly clear for several thousands of years what mathematicians mean when they describe integers as being prime or composite - the concepts and terminology here are neither particularly complex nor in any way disputed. So when someone comes along an says stuff like "2 and 3 are not prime" or "the product of two possible primes is also possibly prime" we have every right to assume they're off their rocker, unless they describe some precise alternative context (say some algebra different than the usual ones people assume when discussing such concepts) in which such statements actually make sense - I've seen nothing of the sort from the author of this thread. If you make on-their-face-outrageous claims it's up to you to justify them, rather than being up to us to try to do so or to read your mind.

Also, many key aspects of Troels' posts precisely match the kinds of criteria which are widely regarded as red flags for crankery:

[b]1) Outrageous-sounding claims;[/b]

[b]2) Use of vague, nonstandard, obfuscatory terminology[/b], when precise, standard, clear wording is readily available to anyone who has studied even the most basic literature in the field;

[b]3) Vague/Hard-to-Find references in place of clear and present argumentation:[/b] Why should I have to hunt up and buy some obscure book that "has been published" by Scientific Vanity Press Inc. (a Division of TrollBooks International) just to try to get even a smidge of the most basic aspects of random-poster-no-one's-ever-heard-of's claims? You think the rest of us have nothing better to do with our time?

I can describe in one or two sentences why e.g. 2 and 3 are prime. Euclid proved in roughly the same amount of space that there are infinitely many primes, starting with just these two (in fact, just one of them suffices, assuming one has the number 1 handy as well). I don't think it's unreasonable to ask for precise justifcation when random-guy comes on the board and posts repeated threads that start with words like "2 and 3 are not possibly prime". Under those circumstances, you're justifiably presumed to be a kook until and unless you prove otherwise.

Your example of Newton v. Leibniz is not applicable here - both of those were undisputedly great, well-established mathematicians, and at the time, Calculus was in its infancy, so wrangling over concepts, terminology and notation were perfectly understandable. I really, really hope you're not saying that you see anything akin to a "Newton" in what we've so far from "Munkner." Newton may have used some pretty bizarre notation (which no one uses anymore - Leibniz et al. clearly won that battle) in his work on Calculus, but he was actually using it to solve highly nontrivial problems and prove interesting results, and thus deserved to be taken seriously. I see nothing remotely analogous here.

[QUOTE=xilman;88463]Minor typo? I think you meant unique [b]factorization[/b].[/QUOTE]
No, I think Bob means "unique multiplication" in the sense of the fundamental theorem of arithmetic, which is thus equivalent to "unique prime factorization".

Bob, please correct me if I misinterpreted you here.

ACKOWLEDGEMENT
 

dear Malcolm,

You are the alone one who has understood my message.
But you happen to have a copy of my original publication.

I realize that I am offending (nearly) all mathematicians,when I say
that we need a new definition of "primes". I have chosen the phrase
"possible primes", which comprise "real primes" and "possible prime products".
It is not possible to obviate the word "--- primes", when you propose
a new concept in this field.

You ask for my reason for challenging 3 as a prime. This integer is in fact
a product of 1 (a real prime) and 3 and is not different from the products
21 or 51 etc., and I consider all odd integers which are divisible by 3
as "never primes" (a terrible misnomer).

I too have challenged Euclid's proof of the infinite number of primes
(in a previous thread) . Euclid has proven that (6*5*7*11 + 1) is not
divisible by 5, 7 or 11, which is correct. But the following steps in his
argument (6*5*7*11*13 +1) are products, and Euclids formulation
becomes the säme as my concept of possible primes (6*m +1). I think
that Euclid has pointed to a number of "possible primes", which comprise
prime products and real primes.

I hope that you will accept my arguments.

Y.s.

troels

[QUOTE=troels munkner;88645]
You ask for my reason for challenging 3 as a prime. This integer is in fact
a product of 1 (a real prime) and 3 and is not different from the products
21 or 51 etc., and I consider all odd integers which are divisible by 3
as "never primes" (a terrible misnomer).[/QUOTE]

troels,

This statement shows why the reputable mathematicians dismiss your argument.

You choose to exclude "3" based on the above argument. However, at the same time, you choose to include "7", et.al.

Just as you argue about "3", one should make the same argument about "7" (and all other integers).

(By your argument) I must question 7 as a prime. This integer is in fact
a product of 1 (a real prime) and 7 and is not different from ...

Similarly, for 11: This integer is in fact a product of 1 (a real prime) and 11 and is not different from ...

Extending that argument, 1 is the only "prime".

The "problem" with your argument is that it is not self-consistent.
Consistency is a REQUIREMENT of mathematics.

The other point that you seem to miss is that this set of numbers really has little to do with primes. There are more members of the set which are non-primes than there are primes. Therefore, your misuse of the term "prime" is objectionable to many of us. May I suggest that you call these numbers "Munkner Numbers"?

Then you can make statements like the following without raising the ire of mathematicians:

All primes > 3 are Munkner Numbers.

The set of Munkner Numbers is closed under multiplication.

etc.

Whether, or not, you can develop any "interesting" results remains to be seen.


Richard

[QUOTE=troels munkner;88645]dear Malcolm,

You are the alone one who has understood my message.
But you happen to have a copy of my original publication.

I realize that I am offending (nearly) all mathematicians,when I say
that we need a new definition of "primes". I have chosen the phrase
"possible primes", which comprise "real primes" and "possible prime products".
It is not possible to obviate the word "--- primes", when you propose
a new concept in this field.

troels[/QUOTE]
:smile:
Thank y0u Troels.
Speaking of terminology here is an extract from the chapter 'Prime Prostitution' by Paul Hoffman where the term 'almost prime' is used which is equally frustrating!

' for almost 150 years, number theorists conjectured that pairs of twin primes are inexhaustible, like the primes themselves, but no one has been able to prove this.

Progress was made in 1966, when the Chinese mathematician Chen Jing-run proved that there exist infinitely many pairs of of numbers that differ by two, in which the first number is a prime and the second is either a prime or the product of two primes. ( A number that is the product of two primes is called
"almost prime", a description that attests to the irrepressible optimism of mathematicians as to the intractability of bona fide prime numbers)

So this is something one has to live with.

"Go on! and Faith will come to you" d'Alembert.

Mally :coffee:

The standard 'taught in elementary school' definition of a prime number is any number that has divisors of only 1 and itself. This definition is broad enough to cover all except one, zero, and negative one for which special cases must be considered.

Your argument is that no number less than 6 can be a prime or maybe it is better stated that only numbers with (6*M) + 1 derived roots can possibly be prime. In effect you seem to have divided the manifold into two areas, numbers from positive to negative infinity which may be prime and numbers from -6 to +6 which seem to exist in some kind of limbo. These two zones are exclusive and do not overlap. Primes may occur only in the - to + infinity range and may not exist in the -6 to +6 range.

It seems to me that a similar argument could be made for all numbers of the form (30*M) + 1 given that 30 is composed of 2*3*5. In effect, I would have a subset of your (6*M) + 1 numbers. My boundaries would be -30 to +30 where yours are -6 to +6.


Fusion

[QUOTE=Fusion_power;88679]The standard 'taught in elementary school' definition of a prime number is any number that has divisors of only 1 and itself. This definition is broad enough to cover all except one, zero, and negative one for which special cases must be considered.

Your argument is that no number less than 6 can be a prime or maybe it is better stated that only numbers with (6*M) + 1 derived roots can possibly be prime. In effect you seem to have divided the manifold into two areas, numbers from positive to negative infinity which may be prime and numbers from -6 to +6 which seem to exist in some kind of limbo. These two zones are exclusive and do not overlap. Primes may occur only in the - to + infinity range and may not exist in the -6 to +6 range.

It seems to me that a similar argument could be made for all numbers of the form (30*M) + 1 given that 30 is composed of 2*3*5. In effect, I would have a subset of your (6*M) + 1 numbers. My boundaries would be -30 to +30 where yours are -6 to +6.


Fusion[/QUOTE]


Please, read my original contribution again.
I said that [(6*m)+1)] with the integer m running from - infinity to + infinity
constitute a special group of integers (exactly one third of all integers).
I happened to call these integers, which are indivisible by 2 and 3,
for possible primes, subdivided into a group of real primes and a group of
real prime products. I don't feel too happy for the the replies, when people
have not understood my message.
I know that I interfere with the antique definition of primes and so be it.

By the way [(6*(-5)+1] = 29, and [(6*(+5)+1) = 31.
Let us put it in another way: "possible primes" are located along a straight line with a difference of 6 between the individual integers
such as --. (-41), (-35), (-29). (-23), (-17), (-11), (-5), 1,7,13,19,25,31, --.
This idea eliminates any further discussion about "twin primes".

Y.s.
troels munkner

Please, read my threads etc. with a kind of open mind.
 

[QUOTE=Fusion_power;88679]The standard 'taught in elementary school' definition of a prime number is any number that has divisors of only 1 and itself. This definition is broad enough to cover all except one, zero, and negative one for which special cases must be considered.

Your argument is that no number less than 6 can be a prime or maybe it is better stated that only numbers with (6*M) + 1 derived roots can possibly be prime. In effect you seem to have divided the manifold into two areas, numbers from positive to negative infinity which may be prime and numbers from -6 to +6 which seem to exist in some kind of limbo. These two zones are exclusive and do not overlap. Primes may occur only in the - to + infinity range and may not exist in the -6 to +6 range.

It seems to me that a similar argument could be made for all numbers of the form (30*M) + 1 given that 30 is composed of 2*3*5. In effect, I would have a subset of your (6*M) + 1 numbers. My boundaries would be -30 to +30 where yours are -6 to +6.


Fusion[/QUOTE]


Please, read my original contribution again.
I said that [(6*m)+1)] with the integer m running from - infinity to + infinity
constitute a special group of integers (exactly one third of all integers).
I happened to call these integers, which are indivisible by 2 and 3,
for possible primes, subdivided into a group of real primes and a group of
real prime products. I don't feel too happy for the the replies, when people
have not understood my message.
I know that I interfere with the antique definition of primes and so be it.

By the way [(6*(-5)+1] = 29, and [(6*(+5)+1) = 31.
Let us put it in another way: "possible primes" are located along a straight line with a difference of 6 between the individual integers
such as --. (-41), (-35), (-29). (-23), (-17), (-11), (-5), 1,7,13,19,25,31, --.
This idea eliminates any further discussion about "twin primes".

Y.s.
troels munkner

[QUOTE=troels munkner;89145]Please, read my original contribution again.
I said that [(6*m)+1)] with the integer m running from - infinity to + infinity
constitute a special group of integers (exactly one third of all integers).[/QUOTE]

(6*m)+1 gives one sixth of all integers. Unless your definition of multiplication and addition are different from mine.

[QUOTE=troels munkner;89145]By the way [(6*(-5)+1] = 29, and [(6*(+5)+1) = 31.[/QUOTE]

And the line above is evidence for it: 6*(-5) is -30 adding one gives -29 not 29.

You are correct, I did mis-understand part of your proposal. However, my underlying concept looks to be intact. Where you are using 2*3, I suggest using 2*3*5. The manifold is divided into groups of numbers and according to your system, only one out of 6 numbers can possibly be prime. My division suggests that only one out of 30 numbers can be prime. (there are a couple of minor quibbles with this, but it is simplest form) Why is your method valid while mine is not? What about 2*3*5*7? It is conceded that none of your numbers can possibly be divisible by 2 or 3. What about numbers of the form (6*M) + 3?

Fusion

[QUOTE=Jacob Visser;89146](6*m)+1 gives one sixth of all integers. Unless your definition of multiplication and addition are different from mine.



And the line above is evidence for it: 6*(-5) is -30 adding one gives -29 not 29.[/QUOTE]



You are absolutely right. A minus sign was missing in "29", which was
a lapsus calami. Later in my text the number was correctly given as (-29).
Sorry for the inconvenience.

Y.s.

troels munkner

[QUOTE=troels munkner;89191]a lapsus calami.[/QUOTE]

I tried writing with squid ink several occasions, but the little buggers are indeed exceedingly slippery, as you note. And training them to produce a steady ink flow rather than just "blurting out" whatever is on their little squiddy minds is ... rather challenging.

Oh wait, I'm thinking of a [i]lapsus calamari.[/i] Sorry about the confusion, ink on my face, & c. Y'all must've thought I was Kraken up mentally, or something.

OK, so I'm just going to ignore the silly and confusing "possible primes" verbiage invented by Mr. Munkner, and instead use simply "integers of the form 6*m+1" wherever it occurs. So now let's look at the substance of the resulting claims:

[QUOTE=troels munkner;88242]"Possible primes" were defined as [(6*m)+1], m being an integer from - infinity to + infinity.
Negative possible primes (-5,-11,-17,-23.....) have modules V, II or VIII.
Positive possible primes (1,7,13,19,25,....) have modules I, VII or IV.[/quote]
So since all primes other than 2 and 3 must be of the form 6*n+-1 (n running over the naturals), basically you catch the ones == +1 (mod 6) with m > 0, and the ones == -1 (mod 6) by way of their arithmetic inverses, using m < 0. This is completely trivial.

[quote]The integers 2 and 3 cannot be defined as possible primes (6*m +1)
and should not be considered as primes.[/quote]
Again ignoring your obfuscatory "possible primes" terminology, this is just the statement that "the primes 2 and 3 are the only ones not of the form 6*m + 1." Again, trivially true.

[quote]The integer 1 is a square (6*0 +1)*(6*0 +1), just as 25 is equal to
[(6*(-1) +1)] * [(6*(-1) +1)] and 49 is equal to [(6*(+1) +1)] * [(6*(+1) +1)].[/quote]
"The square of a number == 1 (mod 6) is == 1 (mod 6)". Trivial, and in fact a subset of another trivial observation:

[quote]Products of possible primes remain possible primes 36 * (n*m) + 6* (n+m) +1, n being an integer from - infinity to +infinity.[/quote]
"The product of two numbers == 1 (mod 6) is == 1 (mod 6)".
Wow - how enlightening.

[quote]All Mersenne primes are positive possible primes and will be defined
in a later thread.[/quote]
Actually, that should read "All Mersenne primes with odd exponent." Since you find the actual proof of this to apparently be quite challenging, allow me to help you out here:

M-primes are of the form M(p) := 2[sup]p[/sup]-1, we consider the ones with p an odd prime, the smallest of which is 2[sup]3[/sup]-1. now observe that 2[sup]3[/sup] == 2 (mod 6), and thus also that 2*2[sup]2[/sup] == 2 (mod 6). It follows that in fact 2*2[sup]2*k[/sup] == 2 (mod 6) for all integter k. Since all odd-prime M(p) must have an exponent of form p = 3+2*k, their power-of-2 component must satisfy 2[sup]p[/sup] = 2[sup]3+2*k[/sup] == 2 (mod 6). QED

[QUOTE=ewmayer;89284]So since all primes other than 2 and 3 must be of the form 6*n+-1 (n running over the naturals), basically you catch the ones == +1 (mod 6) with m > 0, and the ones == -1 (mod 6) by way of their arithmetic inverses, using m < 0. This is completely trivial.[/QUOTE]
Sorry but Troels Munkner uses the form 6*m+1 not (6*m+1 and 6*m-1) that is why I insist that his third of all integers is really a sixth of all integers. A lot of posters here go to the logical conclusion that to have a third of integers (the integers not divisible by 2 nor by 3) one must use (6*m+1 and 6*m-1) but in this thread Troels Munkner uses only 6*m+1. This means divising Z in 4 classes, m being a member of Z:
- the even numbers of the form 6m, 6m+2 and 6m+4;
- the odd multiples of three of the form 6m+3;
- the "Troels Munkner possible primes" or (real primes and possible prime products" of the form 6m+1 and finally
- other numbers of the form 6m+5, the numbers in this last class are not integers by Troels Munkner's definition.
[QUOTE=troels munkner;88444]C. Odd integers which are not divisible by 3.
Their general form is (6*m +1), m being an integer from - infinity to + infinity.
These integers can be grouped as "possible primes" and comprise real primes and possible prime products.
All possible primes are "located" along a straight line wuith an individual difference of 6, i.e. (6*m +1) --- (-35), (-29), (-23), (-17), (-11), (-5),1,7,13,19,25,31,37 --- (6*m +1)
Possible primes constitute exactly one third of allo integers.[/QUOTE]
By this definition 5 could be a possible prime and even a real prime since it is not even and not divisble by three, but since 5 is not of the general form 6m+1 it can not be a real prime nor a product or primes (on this last point I am inclined to think that most will agree with him :-) In fact 5 is not a "Troels Munkner integer". Troels Munkner confirms this here:
[QUOTE=troels munkner;89145]Let us put it in another way: "possible primes" are located along a straight line with a difference of 6 between the individual integers such as --. (-41), (-35), (-29). (-23), (-17), (-11), (-5), 1,7,13,19,25,31, --.[/QUOTE]
Note that a THIRD of all primes are separated by a difference of SIX!
A question for Mally: did you READ Troels Munkners's posts ?

[QUOTE=Jacob Visser;89288]Sorry but Troels Munkner uses the form 6*m+1 not (6*m+1 and 6*m-1) that is why I insist that his third of all integers is really a sixth of all integers.[/QUOTE]

Yes, but as I pointed out (and this is why is used 'n' in my positive-integer expression 6*n+-1, rather than 'm'), he gets the primes == 5 (mod 6) via a "backdoor" route by allowing m to be negative.

My conclusion that there is nothing new or even remotely interesting here stands.

I agree with you that his theorem is just a set of definitions that do not bring new insights.
But I must be dumb: I did not see 5 in his list only -5. And how could 5 and 7 have a difference of 6?
One can see this in his fist port:
[QUOTE=troels munkner;88242]"Possible primes" were defined as [(6*m)+1], m being an integer from - infinity to + infinity.
Negative possible primes (-5,-11,-17,-23.....) have modules V, II or VIII.
Positive possible primes (1,7,13,19,25,....) have modules I, VII or IV.[/QUOTE]
If I read well the modules are modulo 9.

It is true that his arithmetic is a bit shaggy, sometimes one gets the impression that for him [6*(-1)+1] might be equal to [-1*(6*1+1)].

I feel that Mr. Munkner is actually not portraying his Mathematical notations correctly and he is actually meaning to imply the abolute values of ((6*m)+1) is one third of all the positive integers, where m can be either a positive or negative integer. That seems the only way I can make sense of how he has written the notations.

Regards
Patrick

I think the best way to find out what mr. Munkner means would perhaps be asking him plain, simple, easy-to-understand questions.

So, mr. Munkner, please classify both the statements below as either "true" or "false". Whatever your answers are, we've already understood the logic beneath them, so you don't need to spend time explaining why these statements are true or false.

1) The number 5 (positive five) is a prime number. True or false?
2) The number -5 (negative five) is a prime number. True or false?

Please do follow my guidelines strictly, as I personally have a very hard time understanding math which doesn't present itself to me according to them.

Thanks a lot,
Bruno

[QUOTE=Jacob Visser;89288]

Note that a THIRD of all primes are separated by a difference of SIX!
A question for Mally: did you READ Troels Munkners's posts ?[/QUOTE]
:rolleyes:
Well jakes, he very generously sent me his book(let) by airmail which he priced at $15 though he has not given me the bill as yet!

I must confess it is even more confusing than his posts. and not well connected from section to chapter.

Never the less, I personally feel,, that he has a message which he cant explain clearly.

Due paucity of time, as Im working on my own theories myself, I have not really studied it.

But if you are interested I could Xerox the booklet and send it to you by post, just for the asking. You may PM me at your leisure.

Regards,

Mally :coffee:

[QUOTE=mfgoode;89386]:rolleyes:
Well jakes, he very generously sent me his book(let) by airmail which he priced at $15 though he has not given me the bill as yet!

I must confess it is even more confusing than his posts. and not well connected from section to chapter.

Never the less, I personally feel,, that he has a message which he cant explain clearly.

Due paucity of time, as Im working on my own theories myself, I have not really studied it.

But if you are interested I could Xerox the booklet and send it to you by post, just for the asking. You may PM me at your leisure.

Regards,

Mally :coffee:[/QUOTE]
Dear Malcolm,
The book was a gift.
Sorry that you don't have the necessary time yo read the book.
Please, ask me for additional information, and you will get it.
In the near future I will publish a thread about the structure of Mersenne
primes, which follow exactly the same line as the "possible primes".
It is a pity, that other mathematicians don't have an open mind for
new ideas in this field.

Y.s.

troels

[QUOTE=brunoparga;89361]I think the best way to find out what mr. Munkner means would perhaps be asking him plain, simple, easy-to-understand questions.

So, mr. Munkner, please classify both the statements below as either "true" or "false". Whatever your answers are, we've already understood the logic beneath them, so you don't need to spend time explaining why these statements are true or false.

1) The number 5 (positive five) is a prime number. True or false?
2) The number -5 (negative five) is a prime number. True or false?

Please do follow my guidelines strictly, as I personally have a very hard time understanding math which doesn't present itself to me according to them.

Thanks a lot,
Bruno[/QUOTE]



1 is false,
2 is true.

Y.s.
Troels Munkner

[QUOTE=troels munkner;89710]1 is false,
2 is true.

Y.s.
Troels Munkner[/QUOTE]

I ask you then with tears in my baby blue eyes, if postive 5 is not a prime, then according to your definition, it must be composite. Please tell us the factors or is that a bit more difficult to do than factoring RSA-2048??

Patrick123

factors
 

[QUOTE=Patrick123;89717]I ask you then with tears in my baby blue eyes, if postive 5 is not a prime, then according to your definition, it must be composite. Please tell us the factors or is that a bit more difficult to do than factoring RSA-2048??

Patrick123[/QUOTE]
:grin: Now Pat please dont clout me !

I know you are talking about integers and may I ask which integers?

In Gaussian integers indeed 5 has factors and I mention this as its quite a curiosity which struck my fantasy. Here they are;

(1 + 2i )(1 - 2i ) = + 5 and ( 2i + 1 )(2i - 1 ) = -5

Mally :coffee:

[QUOTE=Patrick123;89717]I ask you then with tears in my baby blue eyes, if postive 5 is not a prime, then according to your definition, it must be composite.[/QUOTE]

Patrick - it's clear to me that TM is not using "prime" in the normal sense at all. See my post above, where I wrote

[quote=ewmayer]OK, so I'm just going to ignore the silly and confusing "possible primes" verbiage invented by Mr. Munkner, and instead use simply "integers of the form 6*m+1" wherever it occurs.[/quote]

So when TM says "5 is not prime", just substitute "5 is not of the form 6*m + 1 for integer m", and there you go.

As I also noted, use of misleading/nonstandard/obfuscatory terminology is a hallmark of crankery. If you can't say whatever it is you think you have to say using nonambiguous, easily-understandable standard terminology, that tells me you're either trying to deliberately confuse, or you don't know what you're talking about.

unnecessary tears
 

[QUOTE=Patrick123;89717]I ask you then with tears in my baby blue eyes, if postive 5 is not a prime, then according to your definition, it must be composite. Please tell us the factors or is that a bit more difficult to do than factoring RSA-2048??

Patrick123[/QUOTE]

You don't understand my subdivison of integers into three groups:
a) even integers
b) odd integers divisible by 3 (modules 0,III,VI, modulo 9)
c) odd integers with modules V,II,VIII or I,IV,VII.
These integers can be formulated as [(6*m)+1] with m running
from - infinity to + infinity.
[((6*(-1))+1] = - 5
[(6*1) +1] = 7

Possible primes are "located" along a straigt line of integers
(-----,-35,-29,-23,-17,-11,-5, 1,7,13,19,25,31 -----)

Sorry for your tears. I understand that you don't grasp anything.

Y.s.

troels munkner

[QUOTE=troels munkner;89685]Dear Malcolm,
The book was a gift.
Sorry that you don't have the necessary time yo read the book.
Please, ask me for additional information, and you will get it.
In the near future I will publish a thread about the structure of Mersenne
primes, which follow exactly the same line as the "possible primes".
It is a pity, that other mathematicians don't have an open mind for
new ideas in this field.

Y.s.

troels[/QUOTE]
:smile:
Thank you Troels for the gift. I assure you I will treasure it and when I donate it in my will I hope some math'cian takes it up and completes the theory behind it.

You asked for a proof of the infinitude of primes and you were given Euclid's which is the popular one even in this day

Well there are others which I have but cannot reproduce them due to the complexity of the terms but I will mention their names and you can follow it up from the Book 'The little book of bigger primes' recommended to me by T Rex. and yes it is worth every dollar I paid for it. and the author is Paulo Ribenboim 2nd edition. I'm sure you could locate it in a library near by.

The proofs are by Perrott (1881) Auric (1915), Metrod (1917) and Washington (1980). The last is via commutative algebra.

They have been forgotten and that probably is our fate too in the long run.

Well an Euler or a Gauss turn up every century to redirect the maths path but all cannot claim this honour and neither can they equal them.
So best of luck

Mally :coffee:

[quote="mfgoode"]Gaussian integers ... (1 + 2i )(1 - 2i ) = + 5 and ( 2i + 1 )(2i - 1 ) = -5[/quote]That can't be it, because according to troels munkner -5 is prime.

OK, mr. Munkner, thanks for your answer.

I have another question. You're calling some numbers "prime" and some other "not prime". The former are the ones which fit your 6m+1 formula and the latter are the ones which don't.

My question is, besides fitting the formula above, do primes have any other feature in common? One that fits every prime and no "non-prime"? Please notice that, like my other questions, this one has only two possible answers. The answer must either be "no", or it'll be "yes". If it's "yes" I'd ask you to provide a definition of the common feature of all primes that is as simple, coherent and comprehensive as possible.

Thanks a lot,
Bruno

PS: By common feature I specifically exclude being odd, since that follows from the formula, evidently.

[QUOTE=mfgoode;88454]:rolleyes:
The only reason I can presume to explain this is that 1 is not considered a prime. It is its own square and this property is unique.

Since 2 is considered the only even prime it 'may' also be dropped out of the 'real' prime sequence.

Now Goldbach's conjecture is that every even number greater the two (2=1+1) is the sum of 2 prime numbers. So two is not, by definition above of 1, not being a prime and Goldbach makes 2 an exception to his rule..

Is that what you mean Troels?

But why do you consider 3 as not a prime number? Have you a logical reason?

Mally :coffee:[/QUOTE]


Dear Malcolm,
You have kindly submitted three replies with reference to my thread
"A (new) Prime Theorem".
Please recall my definition of "possible primes" [(6*M)+1], M being any
integer from - infinity to + infinity, zero included. M can simply be called
an integer factor (negative, zero or positive).
Let me give you an example with M = - 10 and M = + 10.
The two possible primes will be - 59 and 61 (by modulation V and VII).
The integer [(6*M)+1] will never be divisible by 2 or 3.

All "possible primes" have modules I,IV,VII or V,II,VIII.
All odd integers divisible by 3 have modules = I,III or VI.

I think that you will understand my proposal for a replacement
of the generally accepted (antique) definition of primes.
It is a pity, that many mathematicians don't understand this new idea.

As a consequence of my change of terminology any discussion of
"twin primes" will be of no avail. At the same time Goldbach's conjecture
will be rejected, as it will be incorrect for the sums 4,6,8,10.

Y.s.

Troels Munkner

Since the words 'prime' and 'possible prime' already have mathematical definitions, we should not change these definitions. These words also have an historical value...

Mr. Munkner, I suggest you to invent a new set of words applying to your definitions.

[QUOTE=troels munkner;89726]You don't understand my subdivison of integers into three groups:
a) even integers
b) odd integers divisible by 3 (modules 0,III,VI, modulo 9)
c) odd integers with modules V,II,VIII or I,IV,VII.
These integers can be formulated as [(6*m)+1] with m running
from - infinity to + infinity.
[((6*(-1))+1] = - 5
[(6*1) +1] = 7

Possible primes are "located" along a straigt line of integers
(-----,-35,-29,-23,-17,-11,-5, 1,7,13,19,25,31 -----)

Sorry for your tears. I understand that you don't grasp anything.

Y.s.

troels munkner[/QUOTE]

Troels, thank you for your understanding where confusion reigns. I have tried to keep an extremely open mind, I think it was so open that my brains fell out. Besides that, if Dr. Bob Silverman, a man who I can only hope to have a fraction of his knowledge, as well as a number of other learned persons on this forum, query your statements, I seriously wonder who is confused:unsure:

I shall however, continue to monitor this thread with a chuckle and snort waiting in bated breath for my epithany!

Regards
Patrick

[QUOTE=troels munkner;89789]


<snip>

[/QUOTE]

I will be patient.....

You have partioned the integers according to their congruence class
mod 6, skipping the class equal to 5 mod 6.

Now what? Do you actually have a theorem to state?

Trivially, the class that is 1 mod 6 is closed under multiplication and
has no member divisible by 2 or 3.

Where do you go from here?

[QUOTE=R.D. Silverman;89800]I will be patient.....[/quote]

I'm starting to get worried about Dr. Silverman ... first he replies politely and with saint-like patience to that insane Raman666 guy that's been stinking up the Factoring forum, now this ... Bob, whatever magic "herb" you recently discovered in your backyard garden - would you consider selling some?

[quote]Where do you go from here?[/QUOTE]

Why, [url=http://en.wikipedia.org/wiki/Up_to_eleven]nowhere.[/url]

[i]"It's one louder, isn't it?"[/i] (Or in this case, "one primer.")

[QUOTE="ewmayer"]Why, [url=http://en.wikipedia.org/wiki/Up_to_eleven]nowhere.[/url][/QUOTE]:bow::goodposting:
That one link has made this whole thread worthwhile.

Eleven
 

[QUOTE=ewmayer;89808]I'm starting to get worried about Dr. Silverman ... first he replies politely and with saint-like patience to that insane Raman666 guy that's been stinking up the Factoring forum, now this ... Bob, whatever magic "herb" you recently discovered in your backyard garden - would you consider selling some?[/QUOTE]
:smile:
It might be the 'magic' herb Brahmi ( Bacopa monnieri ) well known to the Indian gurus. Nature gives the hint as it is shaped like the brain and also has convolutions resembling it.
Ever wondered how the Chinese discovered Ging Seng ? Well the same way thru Nature.

[QUOTE= ] Why, [url=http://en.wikipedia.org/wiki/Up_to_eleven]nowhere.[/url]

[i]"It's one louder, isn't it?"[/i] (Or in this case, "one primer.")[/QUOTE]

That was good reading Ernst. Must check the vol. control on my sons Fender and amp.

That reminds me I hated, decades ago, the sign outside and inside the London pubs. 'We close at Eleven' which prepared the customers to be ready for work the next morning .

On saturdays and sundays we would have 'elevenses'

[url]http://en.wikipedia.org/wiki/Elevenses[/url]

Mally :coffee:

[QUOTE=retina;89811]:bow::goodposting:
That one link has made this whole thread worthwhile.[/QUOTE]

And don't forget to look for our good friend Nigel in one of the recent [url=http://www.youtube.com/watch?v=zwLeRRnZQtU]Volkswagen "V-Dubs Rock" TV Ads[/url]:

[i]"This amplifier has airbags ... (falls over backwards with a loud crash) ... I'm OK..."[/i]

explanation
 

[QUOTE=Wacky;88649]troels,

This statement shows why the reputable mathematicians dismiss your argument.

You choose to exclude "3" based on the above argument. However, at the same time, you choose to include "7", et.al.

Just as you argue about "3", one should make the same argument about "7" (and all other integers).

(By your argument) I must question 7 as a prime. This integer is in fact
a product of 1 (a real prime) and 7 and is not different from ...

Similarly, for 11: This integer is in fact a product of 1 (a real prime) and 11 and is not different from ...

Extending that argument, 1 is the only "prime".

The "problem" with your argument is that it is not self-consistent.
Consistency is a REQUIREMENT of mathematics.

The other point that you seem to miss is that this set of numbers really has little to do with primes. There are more members of the set which are non-primes than there are primes. Therefore, your misuse of the term "prime" is objectionable to many of us. May I suggest that you call these numbers "Munkner Numbers"?

Then you can make statements like the following without raising the ire of mathematicians:

All primes > 3 are Munkner Numbers.

The set of Munkner Numbers is closed under multiplication.

etc.

Whether, or not, you can develop any "interesting" results remains to be seen.


Richard[/QUOTE]


Dear Richard,
Thanks for your comments.
Let me start with your final suggestion. I don't mind to change terminology
from "possible primes" to "Munkner integers". These expressions cover in fact
the same integers. To use "numbers" instead of "integers" makes no difference, but points to the fact (already mentioned in my publicationn
from 1986) that all "Munkner integers" can be "replaced" by their natural
number M from - infinity to + infinity:--(-7),(-6),(-5),(-4),(-3),(-2),(-1),
0,1,2,3,4,5,6 --, corresponding to (-41),(-35),(-29),(-23),(-17),(-11),(-5),
1,7,13,19,25,31,---
Now I think that you agree to my statement that ((6*M)+1) will never be
divisible by 2 or 3, as (6*M) is divisible by 2 and 3.
Odd integers with modules 0,III or VI (modulo 9) will never be primes
(21 has module III, 51 has module VI, 117 has module 0).
The rest of the odd integers with modules II,V,VIII or I,IV,VII will be
primes or prime products (i.e. "Munkner integers").
Your remark "all primes > 3 are Munkner numbers" should read
"all primes > 3 and all prime products of primes > 3 are Munkner integers".
But you forget the integer 1, which is a product (a square)
of ((6*0)+1) and ((6*0)+1).
You may ask me about the advantages of the terminology "Munkner integers".
They will be evident when I describe the dissection (i.e. the factorization)
of Munkner integers and in addition the dissection of the "Mersenne integers"
which constitute a few primes and a vast majority of prime products.

Y.s. troels

[QUOTE=troels munkner;89870]Dear Richard,
Thanks for your comments.
Let me start with your final suggestion. I don't mind to change terminology
from "possible primes" to "Munkner integers". These expressions cover in fact
the same integers. To use "numbers" instead of "integers" makes no difference, but points to the fact (already mentioned in my publicationn
from 1986) that all "Munkner integers" can be "replaced" by their natural
number M from - infinity to + infinity:--(-7),(-6),(-5),(-4),(-3),(-2),(-1),
0,1,2,3,4,5,6 --, corresponding to (-41),(-35),(-29),(-23),(-17),(-11),(-5),
1,7,13,19,25,31,---
Now I think that you agree to my statement that ((6*M)+1) will never be
divisible by 2 or 3, as (6*M) is divisible by 2 and 3.
Odd integers with modules 0,III or VI (modulo 9) will never be primes
(21 has module III, 51 has module VI, 117 has module 0).
The rest of the odd integers with modules II,V,VIII or I,IV,VII will be
primes or prime products (i.e. "Munkner integers").
Your remark "all primes > 3 are Munkner numbers" should read
"all primes > 3 and all prime products of primes > 3 are Munkner integers".
But you forget the integer 1, which is a product (a square)
of ((6*0)+1) and ((6*0)+1).
You may ask me about the advantages of the terminology "Munkner integers".
They will be evident when I describe the dissection (i.e. the factorization)
of Munkner integers and in addition the dissection of the "Mersenne integers"
which constitute a few primes and a vast majority of prime products.

Y.s. troels[/QUOTE]


(1) Only a crank names a mathematical idea after himself.
(2) You still have not said anything intelligent.

[quote=R.D. Silverman;89872](1) Only a crank names a mathematical idea after himself.[/quote]Like Mersenne, Fermat, Fibonacci, Sophie Germain, and countless others?

[QUOTE=R.D. Silverman;89872](1) Only a crank names a mathematical idea after himself.
(2) You still have not said anything intelligent.[/QUOTE]To be fair to someone who shows most of the classical signs of being a crank, it was Richard Wackerbarth who wrote: 'May I suggest that you call these numbers "Munkner Numbers"?'

I applaud the change in nomenclature. It greatly reduces opportunities for confusion.

Like you, I'm still waiting for something intelligent to be said.

Paul

[QUOTE=Mini-Geek;89873]Like Mersenne, Fermat, Fibonacci, Sophie Germain, and countless others?[/QUOTE]

These names were assigned by others. The inventors did not name
them after themselves.

In a similar nomenclatural vein, I propose that all integers divisible by 11 be named "[url=http://en.wikipedia.org/wiki/Nigel_Tufnel]Tufnel[/url] Numbers."

To be safe - one never knows where modern electronic amplifier technology will take us - I'll add positive infinity to that lot, as well.

It's a fascinating fact (let's just call it "Mayer's Super-Great Observation") that the smallest positive Tufnel Number is also the smallest prime p such that 2[sup]p[/sup]-1 is not prime. It's "one more divisible", isn't it?

[QUOTE=ewmayer;89901]It's a fascinating fact (let's just call it "Mayer's Super-Great Observation") that the smallest positive Tufnel Number is also the smallest prime p such that 2[sup]p[/sup]-1 is not prime. It's "one more divisible", isn't it?[/QUOTE]

And because it isn't true for 3[sup]p[/sup]-1, let's call its property : one more but not two "much" divisibility... astounding ! :grin:

A (newer) prime theorem
 

Muddling through this thread provided great amusement and great insights into prime numbers. The former caused me to snort cold beverage through my nose; I'll share the latter.

In the vein of t.m., we can subset all integers. The first such grouping includes all integers less than 10. As manly mathematicians, interested in only the most masculine manifestations of modern methodology, we term these "puny numbers". As such, we have no use for them and refuse to consider them further.

The second subset of integers contains all values greater than or equal to 20. If we consume that many beers in one day, we feel truly terrible the next morning. Hence we fear these numbers and eliminate them from the realm of possibility.

We therefore are left with only four prime numbers: 11, 13, 17, and 19. As you can see, we have not only proven there are finitely many primes (take [i]that[/i] Euclid, Goldbach, and even the good Mr. Furstenberg!), but we have listed each and every one. That is all ye know on earth, and all ye need to know. Searching for other primes is either vanity or foolishness.

[QUOTE=ewmayer;89901]It's a fascinating fact (let's just call it "Mayer's Super-Great Observation") that the smallest positive Tufnel Number is also the smallest prime p such that 2[sup]p[/sup]-1 is not prime. It's "one more divisible", isn't it?[/QUOTE]

In fact, for t a Tufnel number and n an integer, n^t-1 is never prime.

Heh. As far as the Munker integers go, early on in the thread I thought I could make them sensible by defining equivilence classes (defined by absolute values) over the integers, so that |5|=|-5| is prime, and that Munker integers could contain all odd primes not divisible by 3, considered as equivilence classes over the integers. Alas, it was not to be so.

Troels, a question: What properties do these numbers have that makes them interesting? Ordinary integers can't be uniquely factored into Munker integers, and you haven't proposed a use for them that I've read. Are they supposed to be a 'first step' in finding primes? In that case they're a specific case of the wheel sieve with 2 and 3 as spokes.

Patterns of primes
 

I was schocked to read troels munker's posting on "a (new) priome theorem". I have doing some study on expressing numbers in different bases. One direction this has taken was looking at the last digit of numbers in different bases, especially primes. I had been beating my head against a wall since the late 1980's trying to find if I was onto something new.

The proper claim is all prime numbers, LARGER THAN SIX, can be described by 6n+1 or 6n-1. This claim only works one way. Not all numbers equal to 6n+1 or 6n-1 are primes, hence when I used the phrase "possible prime" it meant further study was needed to varify if a number was prime. If ewmayer knows a better term I would like to see it.

I am curious to know if troels munker was ever at Pan American University at Brownsville, which due to mergers is now University of Texas, Brownsville. The number theory experts there were unable to say if there was anything new to what I was doing, or give any advice on who would be better to contact. In fact since the work was based on other number bases, they thought I needed to prove a prime expressed in a different number base was still prime, as if a prime number would grow factors when you converted it from base ten to base six. Since they could not give any advice on what kind of format to use to write up my work it has never been published. It would be very ineresting to find whether I was independently duplicating the work he cites, or if there are other phrases i coined for what I was doing are in this work.

To prove the claim, All primes larger than six must be 6n+1 or 6n-1, consider:
Any number N, expressed in a base B will have a form of Bn+d where d is the digit in the units position. if n>B, then n=iB+j and N will be ijd in base B or iBB+jB+d. Since, for this proof, qwe are only concerned with the units digit Bn+d is sufficient.
In base six N will be 6n+d. d has six values 0,1,2,3,4,5.
When d=0, 6n+d=6n since 6 is a composite number, 6n will be a composite number and not prime.
When d=2, 6n+d= 6n+2=2(3n+1) which will always be a multiple of two and not prime.
When d=3, 6n+d=6n+3=3(2n+1) which will always be a multiple of three and not a prime.
When d=4, 6n+d=6n+4=2(3n+2) which will again always be a multiple of two and not prime.
This leaves d=1 and d=5 which can not be factored this way. This means all primes which are multi-digit in base six, or more simply large that six, must end a 1 or a 5 in base six. (5 = 6-1 so we can say 6n-1 so long as the quanity (6n-1) larger than six.) Converting this back to base ten we can say prime numbers larger than six must be 6n+1 or 6n-1, but not all 6n+1 or 6n-1 are primes.

With regard to one third of all numbers being possible primes it would be more clear to say of any six consecutive numbers large than six there will be no more than two primes, possibly less.

By the same kind of analysis I can claim all primes larger that thirty will fit 30n+1, 30n-1, 30n+7, 30n-7, 30n+11, 30n-11, 30n+13, or 30n-13. Which also means of any thirty consecutive greater than thirty no more than eight will be prime.

[QUOTE=Terence Schraut;89996]I was schocked to read troels munker's posting on "a (new) priome theorem". I have doing some study on expressing numbers in different bases. One direction this has taken was looking at the last digit of numbers in different bases, especially primes. I had been beating my head against a wall since the late 1980's trying to find if I was onto something new.

The proper claim is all prime numbers, LARGER THAN SIX, can be described by 6n+1 or 6n-1. This claim only works one way. Not all numbers equal to 6n+1 or 6n-1 are primes, hence when I used the phrase "possible prime" it meant further study was needed to varify if a number was prime. If ewmayer knows a better term I would like to see it.

I am curious to know if troels munker was ever at Pan American University at Brownsville, which due to mergers is now University of Texas, Brownsville. The number theory experts there were unable to say if there was anything new to what I was doing, or give any advice on who would be better to contact. In fact since the work was based on other number bases, they thought I needed to prove a prime expressed in a different number base was still prime, as if a prime number would grow factors when you converted it from base ten to base six. Since they could not give any advice on what kind of format to use to write up my work it has never been published. It would be very ineresting to find whether I was independently duplicating the work he cites, or if there are other phrases i coined for what I was doing are in this work.

To prove the claim, All primes larger than six must be 6n+1 or 6n-1, consider:
Any number N, expressed in a base B will have a form of Bn+d where d is the digit in the units position. if n>B, then n=iB+j and N will be ijd in base B or iBB+jB+d. Since, for this proof, qwe are only concerned with the units digit Bn+d is sufficient.
In base six N will be 6n+d. d has six values 0,1,2,3,4,5.
When d=0, 6n+d=6n since 6 is a composite number, 6n will be a composite number and not prime.
When d=2, 6n+d= 6n+2=2(3n+1) which will always be a multiple of two and not prime.
When d=3, 6n+d=6n+3=3(2n+1) which will always be a multiple of three and not a prime.
When d=4, 6n+d=6n+4=2(3n+2) which will again always be a multiple of two and not prime.
This leaves d=1 and d=5 which can not be factored this way. This means all primes which are multi-digit in base six, or more simply large that six, must end a 1 or a 5 in base six. (5 = 6-1 so we can say 6n-1 so long as the quanity (6n-1) larger than six.) Converting this back to base ten we can say prime numbers larger than six must be 6n+1 or 6n-1, but not all 6n+1 or 6n-1 are primes.

With regard to one third of all numbers being possible primes it would be more clear to say of any six consecutive numbers large than six there will be no more than two primes, possibly less.

By the same kind of analysis I can claim all primes larger that thirty will fit 30n+1, 30n-1, 30n+7, 30n-7, 30n+11, 30n-11, 30n+13, or 30n-13. Which also means of any thirty consecutive greater than thirty no more than eight will be prime.[/QUOTE]True, but almost completely trivial. You were, quite literally, reinventing the wheel.

Try researching the word "wheel" as used in the context of primality testing and searching. Learning more about the phrase "admissible constellation" may also prove enlightening.


Paul.

Ha, ha, looks like now we're even gonna have a priority dispute...

p.s.: To those of you who haven't seen it (either because you simply don't watch enough TV or live outside the U.S.), I've added a YouTube link to a video clip of the Nigel Tufnel VW Ad in my [url=http://www.mersenneforum.org/showpost.php?p=89820&postcount=50]post above[/url]. Enjoy...

[QUOTE=CRGreathouse;89983]Troels, a question: What properties do these numbers have that makes them interesting? -snip- In that case they're a specific case of the wheel sieve with 2 and 3 as spokes.[/QUOTE]

This is exactly what I was telling myself : a very long thread for the first step of the wheel sieve....:mally:

answer
 

[QUOTE=CRGreathouse;89983]In fact, for t a Tufnel number and n an integer, n^t-1 is never prime.

Heh. As far as the Munker integers go, early on in the thread I thought I could make them sensible by defining equivilence classes (defined by absolute values) over the integers, so that |5|=|-5| is prime, and that Munker integers could contain all odd primes not divisible by 3, considered as equivilence classes over the integers. Alas, it was not to be so.

Troels, a question: What properties do these numbers have that makes them interesting? Ordinary integers can't be uniquely factored into Munker integers, and you haven't proposed a use for them that I've read. Are they supposed to be a 'first step' in finding primes? In that case they're a specific case of the wheel sieve with 2 and 3 as spokes.[/QUOTE]


answer:
6 times all natural numbers (M) from - infinity to + infinity +1
will be an integer of the form (6*M +1), which will never be divisible by
2 or 3. In other words ((6*M)+1) comprise all primes and prime products.
Please, read my reply to Wacky.
As to the use of these integers: ((6*M)+1) *((6*N)+1)=
36*(M*N) + 6*(M+N) +1. If you can find the roots to the second order
equation ([ M+N] +/- (SQRT ((M+N)^2 - 4*M*N))/2 (and that is not very
difficult) you have proved that the integer is a prime product, if you only
(only !) find one root (i.e. N=0) you have proved that the integer is a prime.
If necessary ask for more information.
A dissection of the "Mersenne integers" will follow similar lines, to be published. I consider Mersenneforum for the right place for an analysis
of the "Mersenne primes".
Y.s. troels

[QUOTE=troels munkner;90058]As to the use of these integers: ((6*M)+1) *((6*N)+1)= 36*(M*N) + 6*(M+N) +1. If you can find the roots to the second order equation ([ M+N] +/- (SQRT ((M+N)^2 - 4*M*N))/2 (and that is not very difficult) you have proved that the integer is a prime product, if you only
(only !) find one root (i.e. N=0) you have proved that the integer is a prime.
If necessary ask for more information.[/QUOTE]
Beautiful!!! This is where Euler, Gauss, Riemman could go no further but Troels Munkner could: if one finds a number M that satisfies ((6*M)+1) *((6*N)+1)= 36*(M*N) + 6*(M+N) +1 with N=0 it is a prime!!! I.e. any number ((6*M)+1) *((6*0)+1)= 36*(M*0) + 6*(M+0) +1=(6*M)+1 is a prime!!! 25, 49, 55, for instance.

Or did Mr. Munkner mean M=N instead of N=0 (in whiche case the equation ([ M+N] +/- (SQRT ((M+N)^2 - 4*M*N))/2=(M^2+/-(SQRT((2M)^2-4M^2))/2=M^2/2)? Or something else still?

Why not redefine all positive integers as follows:

1, 2, 3, 5, & 7 are "fundemental" numbers
any number >9 that is divisible by only by itself and 1 would be a prime by the new def.

This makes as much or more sense than what was proposed.

For other bases, fundemental numbers would be those that indivisible and are only 1 single character long. So in base 16, B and D are prime.

[QUOTE=troels munkner;90058]answer:
6 times all natural numbers (M) from - infinity to + infinity +1
will be an integer of the form (6*M +1), which will never be divisible by
2 or 3. [/QUOTE]

Moron. 6 * product (i= -oo to oo) of (6i+1) DOES NOT CONVERGE.
IT IS NOT AN INTEGER. Basic Calculus.

Would someone please get rid of this crank?

First Principles
 

[QUOTE=R.D. Silverman;90072]Moron. 6 * product (i= -oo to oo) of (6i+1) DOES NOT CONVERGE.
IT IS NOT AN INTEGER. Basic Calculus.

Would someone please get rid of this crank?[/QUOTE]

:smile:
With all the criticisms directed to you, Troels, I know I am throwing a straw to a drowning man. So please dont use off the cuff statements without testing them out, either this, or face ridicule.

From First Principles.

In these and subsequent posts in this thread I’m taking on the role of Marin Mersenne, who kept topics of maths alive between several math’cians and their theories, in an open correspondence with one and all.

As such the maths presented here are not mine but those of Troels Munkner from his book ‘A Prime number Theorem’. Kindly address all queries , valuable hints, criticism to him and not to me.

Troels, allow me to take relevant extracts from the above mentioned book and condense them systematically, as Your book is not as coherent as it should be.

I hope you will clarify many questions that may be raised so that your theory may be fully and significantly discussed and analysed, and further developed.

All posters and partakers kindly treat this as a GIMPS project to get a better knowledge of Primes and perhaps an appropriate and faster algorithm than what is in use, and I say Perhaps !

Munkner starts with the classification of integers in his opening chapter.

1) The Integers: All integers have been subdivided into two sets of numbers called (1) ‘Never primes’ (NP) and (2) Possible primes (PP)

2) The Never Primes: These comprise all even numbers AND all odd numbers divisible by 3
On the number line NP are located symmetrically around 0 and so may be called 0-centrred integers.
NP constitute 2/3 of all numbers including two real primes No.s 2 and 3.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers.
PP can be subdivided into real primes and Prime Products (P’P’)
PP constitute 1/3 of all numbers.
PP constitute a multiplicative system which becomes ‘skew’ because of the off set of one in the regularity of factors.
It is not easy to find the prime factors except when the number is a square number, you substitute the PP by 1 x the (Number in question).

Munkner goes on to give a graphical display of this series by sine curves.

Even Integers: 2. sin (pi.N/2)

Never Primes: divisible by 3 as 6 sin (pi.N/6 + pi/2) .

For possible Primes 5 , 7 , 11 and 13 …

30. sin (pi.N./30 + pi/6 ) ; 42 sin ( pi. N/42 – pi/6)
66.sin (pi.N/66 + pi/6) ; 78. sin (pi.N/78 – pi/6 )

Then he gives a composite graphical display of all the curves

His next chapter will be on Prime products and I reserve this for another post

.I welcome your comments and analysis and refrain from my own.

Mally :coffee:

[quote=mfgoode;90075]...I know I am throwing a straw to a drowning man...[/quote]

Throwing a straw? What is a drowning man going to do with a straw?

:mike:

[COLOR=White]along with alone caseztuchz in ni
to express everything out of fo r those the is impossible! inpossible ing ion outy uoty withy wihty iny niy outhy uothy outh uoth[/COLOR]

A Straw.
 

[QUOTE=Xyzzy;90076]Throwing a straw? What is a drowning man going to do with a straw?

:mike:

[COLOR=White]along with alone caseztuchz in ni
to express everything out of fo r those the is impossible! inpossible ing ion outy uoty withy wihty iny niy outhy uothy outh uoth[/COLOR][/QUOTE]
:wink:
It means I am not of much help but I offer hope. :rolleyes:

Mally :coffee:

[QUOTE=Xyzzy;90076]along with alone caseztuchz in ni
to express everything out of fo r those the is impossible! inpossible ing ion outy uoty withy wihty iny niy outhy uothy outh uoth[/QUOTE]
Uh oh - looks like someone's been spending too much time in the Factoring forum...

[QUOTE=Xyzzy;90076]along with alone caseztuchz in ni
to express everything out of fo r those the is impossible! inpossible ing ion outy uoty withy wihty iny niy outhy uothy outh uoth[/QUOTE]Could this be a forum bug? Or perhaps malware on Xyzzy's PC? :confused:

[QUOTE=Uncwilly;90067]For other bases, fundemental numbers would be those that indivisible and are only 1 single character long. So in base 16, B and D are prime.[/QUOTE] Correction:
B and D in base 16 would be fundamentals, but 11 and 13 would not be in base 10.

[QUOTE=retina;90093]Could this be a forum bug? Or perhaps malware on Xyzzy's PC? :confused:[/QUOTE]

Oh, it's forum bug (or, thankfully, now ex-forum bug) alright, but not in the way you're thinking. I think Xyzzy is just having a bit of fun using the peculiar syntactical style of that now-banned Raman666 nutter who started that massive GMP_ECM thread (more of a personal insanity blog, actually) on the Factoring forum. Actually, it *would* be kinda cool to have a feature that would allow one to Raman-ize selected posts - but I asked Xyzzy about an analogous pig-latinizing feature last April 1st, and he said it was a no-go, so if you want your posts to look "special" this way, you ottagay oday itay ourselfyay.

[QUOTE=xilman;88448]This statement is both correct and trivial. It also defines the term "possible prime" to mean integers of the form [tex]6k\pm1[/tex]. No-one but you uses the term "possible prime" but that's ok.

Where you go seriously off the rails is your claim that neither 2 nor 3 is a prime number. By making this statement you are not using the word "prime" in the same sense as it is used by essentially all mathematicians.

Paul[/QUOTE]

Since this forum is so popular (I have no idea as to why), I thought I might correct the mistake he made in saying the statement is correct. Troels defined "possible prime" as [tex]6k+1[/tex].

Possible prime.
 

[QUOTE=fetofs;90222]Since this forum is so popular (I have no idea as to why), I thought I might correct the mistake he made in saying the statement is correct. Troels defined "possible prime" as [tex]6k+1[/tex].[/QUOTE]
:smile:
I reiterate Troels definitions.

2) The Never Primes: These comprise all even numbers AND all odd numbers divisible by 3
On the number line NP are located symmetrically around 0 and so may be called 0-centrred integers.
NP constitute 2/3 of all numbers including two real primes No.s 2 and 3.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers.
Mally :coffee:

Paterns of primes, part 2
 

Well, background always seems trivial.

A prime number is always a prime number. If prime numbers of a specified size can only end in certain digits in a given base, than they can not end in any other digit in that base. Therefore, if a number when expressed as a multi-digit number in some base does not end in a digit available for multi-digit prime numbers in that base, then it is not prime.

Every base has its own set of digits of which a multi-digit prime must end. For example:
Base six multi-digit primes must end in 1 or 5
Base ten multi-digit primes must end in 1, 3, 7, or 9
Base twelve multi-digit primes must end in (digit equivelents of) 1, 5, 7, or 11
Base fifteen multi-digit primes must end in (digit equivelents of) 1, 2, 4, 7, 8, 11, 13, or 14
Base thirty multi-digit primes must end in (digit equivelents of) 1, 7, 11, 13, 17, 19, 23, or 29
Base two hundred ten multi-digit primes must end in (digit equivelents of) 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, or 209.

Look at these as paterns or cycles. Of the six listed above six, ten, fifteen, thirty, and two huindred ten are distinct from each other. As far as spacing of prime numbers is concerned the cycle for base twelve is two repeations of the cycle for base six.
Consider these cycles the way we treat other waveforms. They can behave similarly to constructive and destructive interference. A prime number would occur whenever the cycles for all bases, less than the number, positively reinforce. If any base, where a number is multi-digit, it ends in a not available digit (for example 12 in base thirty, which would be a multiple of six.) than that number is not prime.

[QUOTE=Terence Schraut;90392]Every base has its own set of digits of which a multi-digit prime must end.[/QUOTE]
Your observations about ending digit are trivial and well-known. The explanation:
If a number n ends in digit d when written in base b, then the number has the form n = k*b+d. If a prime p divides both b and d, then p also divides n. This means the only chance of multi-digit primes is when b and d are relatively prime. Dirichlet's theorem says there are infinitely many primes in all such cases. It has also been proved that each d which is relatively prime to n produces the same number of primes asymptotically.

[QUOTE=mfgoode;90227]:smile:
I reiterate Troels definitions.

2) The Never Primes: These comprise all even numbers AND all odd numbers divisible by 3
On the number line NP are located symmetrically around 0 and so may be called 0-centrred integers.
NP constitute 2/3 of all numbers including two real primes No.s 2 and 3.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers.
Mally :coffee:[/QUOTE]

Dear Malcolm,
Thanks for your replies to other mathematicians and to me.
I have used +1 as the centre for all primes and prime products, and it
has a number of advantages.
The expression ((6*M)+1) comprises all primes and prime products,
M being any or all of the natural numbers from - infinity to + infinity.
((6*(-39)+1) = -233, which´is a prime
((6*(+39)+1) = 235, which is a prime product.

A prime product such as ((6*M)+1) * ((6*N)+1) = 36 (NM) + 6*(M+N) + 1
is an integer, which will never be divisible by 2 or 3. Conclusion: 2 and 3
could be called anything but "primes".

N (just as M) being any or all natural numbers from - infinity to + infinity.
(+) * (+) is of course (+), (-) * (-) will also give a (+) integer,
(+) * (-) will give a (-) integer.
All primes and prime products are divisible by 1 (i.e. N=0).
If you want to look for (M) and (N), which means to factorize a possible
prime, you can do it by subtracting 1 from the integer in question and then
use a second order equation to find or not find the two roots (M) and (N).
If the sum of (M+N) is odd and > 1, the factorization results in
(Even integer)^2 - 3^2 * (an odd integer)^2.
If the sum of (M+N) is 0 or any other even number, the factorization ends in
(Odd integer)^2 - 6^2 * (any integer, including 0)^2.

The sign of a prime or prime product can easily be predicted by modulation
(modulo 9), and it is easy to show if the sum (M+N) is even or odd.

If you have the time you can try to follow my ideas:
7*13 = 91 = 10^2-3^2 etc.
7*19 = 133 = 13^2-6^2 etc.

I am not drowning. I will in fact consider to reflect to the many harsh replies,
which I have received (directly or indirerctly), but maybe it is not worth
the effort. A famous citation from Schiller's Jeanne d'Arc comes to my mind
("-----").

Y.s.
troels

[QUOTE=troels munkner;90444]Dear Malcolm,
Thanks for your replies to other mathematicians and to me.
I have used +1 as the centre for all primes and prime products, and it
has a number of advantages.
The expression ((6*M)+1) comprises all primes and prime products,
M being any or all of the natural numbers from - infinity to + infinity.
((6*(-39)+1) = -233, which´is a prime
((6*(+39)+1) = 235, which is a prime product.

A prime product such as ((6*M)+1) * ((6*N)+1) = 36 (NM) + 6*(M+N) + 1
is an integer, which will never be divisible by 2 or 3. Conclusion: 2 and 3
could be called anything but "primes".

<snip>

[/QUOTE]

I must confess to a personal failing.

I do not understandand how people can be so totally clueless as
to spew the kind of nonsense that has been spewed by this troll.

The sad part is that he isn't even aware of how totally clueless
his posts have been.

[QUOTE=R.D. Silverman;90451]I must confess to a personal failing.

I do not understandand how people can be so totally clueless as
to spew the kind of nonsense that has been spewed by this troll.

The sad part is that he isn't even aware of how totally clueless
his posts have been.[/QUOTE]

Chortle, chortle.... snigger, snigger... I hearby declare 6(now named a Monkey Prime) a prime number as no other prime number(according to Munker) is a factor of it:lol:

Regards
Patrick

Just as the song goes,"Everybody plays the fool...No exception to the rule..."

EVERYBODY looks stupid at some point, whether they're simply wrong or misunderstood.

I'm not attempting a threat in any way, but I would advise people to not have an overly large amount of fun at Mr. Munkner's expense. As someone who tries to stay in tune with the Holy Spirit, I know that sometimes this stuff can pop up again and give us an unpleasant view of ourselves, something we would rather not aknowledge about ourselves.

As I reread the above, I realize what I said might not even make sense to Christians, so I'll rephrase it: Sometimes when we judge something unfairly, we can suffer for it later on.

Kindly clarify.
 

[QUOTE=troels munkner;90444]Dear Malcolm,
Thanks for your replies to other mathematicians and to me.
I have used +1 as the centre for all primes and prime products, and it
has a number of advantages.[/QUOTE]

Troels Please clarify this statement. If you put +1 as the centre on the number line does this be like zero on the normal number line? For instance how would you place the prime 7 on this line? Will it be 6 units away from the centre 1 ? or what ?

What most have been confused about is what are the factors of 6.?
We know in real numbers that these are 2 x 3. If you dont include these as prime factors according to your definition, as 2 is an even number and 3 is divisible by 3, and these you call 'never primes', then what would you just call them ?

[QUOTE =Troels} I am not drowning. I will in fact consider to reflect to the many harsh replies,
which I have received (directly or indirerctly), but maybe it is not worth
the effort. A famous citation from Schiller's Jeanne d'Arc comes to my mind
("-----").[/QUOTE]

:sad:
You have mistaken my sentence which I clarified to Mike also. I have not meant that YOU are a drowning man but I can only offer you as much as a drowning man would feel if I threw him a straw. In simple words ' I cannot offer much help in your theory'.

I will however endeavour to bring out what you mean and put it more coherently for others to understand

Regards

Mally :coffee:

Dear Malcolm,

Thanks for your clarification. The expression ((6*M)+1) comprises all
primes and prime products, M being any natural number from - infinity
to + infinity. But these integers will never be divisible by 2 or 3.

Rather soon you will see some new (and important) threads from me.
I will appreciate your comments.

Y.s.
troels

[QUOTE=ewmayer;88324]If that is true, and APNT is basically the same bogus crap you keep posting to this board every few months, it seems their editorial standards are "colossally" low.


You have a most curious definition of "possible prime." Note that I do not intend "curious" to imply in any way that your "definition" is interesting - rather, the kinds of descriptive terms that do come to mind include "idiotic," "clueless", and "wasteful of other people's time." Might I suggest that you either keep your inane musings to yourself, or take them elsewhere? I normally would refrain from using such harsh language, but this is not the first time you've posted this garbage here.


And this tells us what, exactly? That certain perfect squares are ... perfect squares?


So the primes 2 and 3 are in fact not prime, at the same time that any product of your designated 6k+1 possible primes is not clearly composite? You sir, are a moron.


I tremble at the thought of the further "enlightenment" you speak of.[/QUOTE]


Whoever you are I can predict that you (sooner or later) will regret some of
the replies, which you (directly or indirectly) have sent to me.
From your comments I realize that you have limited knowledge of Latin
and have not looked into a dictionary with translations of foreign phrases.
If you prefer modern, more technical expressions in Esperanto or alike,
the translation of "lapsus calami" will be a "typo".

It is not worthwhile to react to your other replies.

Try to read my threads or replies, open-minded for new ideas.
When you pretty soon will see new threads on "Fermat's small theorem",
"An analysis of (the very few) Mersenne primes and the vast majority of
[2^p -1] products", a new tool "SCET (an acronym for Smallest Common
Exponential Term, radix 2)" and "Riemann's zeta-function" and maybe
want to open these threads, please swallow a couple of tranquillizers before
you make your comments.

Perhaps you can find a translation (in your dictionary) of the following quotation: "Quousque tandem abutere patientia".

Y.s.
troels munkner

[QUOTE=ewmayer;88324]If that is true, and APNT is basically the same bogus crap you keep posting to this board every few months, it seems their editorial standards are "colossally" low.


You have a most curious definition of "possible prime." Note that I do not intend "curious" to imply in any way that your "definition" is interesting - rather, the kinds of descriptive terms that do come to mind include "idiotic," "clueless", and "wasteful of other people's time." Might I suggest that you either keep your inane musings to yourself, or take them elsewhere? I normally would refrain from using such harsh language, but this is not the first time you've posted this garbage here.


And this tells us what, exactly? That certain perfect squares are ... perfect squares?


So the primes 2 and 3 are in fact not prime, at the same time that any product of your designated 6k+1 possible primes is not clearly composite? You sir, are a moron.


I tremble at the thought of the further "enlightenment" you speak of.[/QUOTE]


Whoever you are I can predict that you (sooner or later) will regret some of
the replies, which you (directly or indirectly) have sent to me.
From your comments I realize that you have limited knowledge of Latin
and have not looked into a dictionary with translations of foreign phrases.
If you prefer modern, more technical expressions in Esperanto or alike,
the translation of "lapsus calami" will be a "typo".

It is not worthwhile to react to your other replies.

Try to read my threads or replies, open-minded for new ideas.
When you pretty soon will see new threads on "Fermat's small theorem",
"An analysis of (the very few) Mersenne primes and the vast majority of
[2^p -1] products", a new tool "SCET (an acronym for Smallest Common
Exponential Term, radix 2)" and "Riemann's zeta-function" and maybe
want to open these threads, please swallow a couple of tranquillizers before
you make your comments.

Perhaps you can find a translation (in your dictionary) of the following quotation: "Quousque tandem abutere patientia".

Y.s.
troels munkner

Temporary grammar nazism
 

It's interesting that Troels criticizes someone else's Latin knowledge and makes such an elementary mistake in the same post.

The correct sentence would be "quousque tandem abutere patientia[B]m[/B]". Interestly, it applies perfectly to you, mr. Munkner; how long are you gonna keep coming here, trying to persuade everyone that 2 and 3 aren't prime and that Euclid was a moron?

Bruno

[QUOTE=troels munkner;88444]a polite dialogue will be appreciated
All integers from - infinity to + infinity can be subdivided into three groups.
A. Even integers which will be products of 2 and an other integer.
B. Odd integers divisible by 3 which will be products of 3 and an other
odd integer.
C. Odd integers which are not divisible by 3.
Their general form is (6*m +1), m being an integer from - infinity
to + infinity.
[/QUOTE]

Dear troels

I followed this thread for quite a while now and I always asked myself a very simple question: To which of your three groups do the integer numbers 5, 11, 17, 23, ... belong?

I hope you can enlighten me.

Rde:

If I understand that part of troels "terminology" correctly (which I may not), numbers of the form (6*(+m) - 1) are constructed as -(6*(-m) + 1). Like this:
5 -> 6*(-1) + 1 = -5
11 -> 6*(-2) + 1 = -11
17 -> 6*(-3) + 1 = -17
23 -> 6*(-4) + 1 = -23

This is what he means by using +1 as the centre for primes and prime products. If he chose -1, then 5 = 6*1 - 1, and 7 = 6*(-1) - 1. That is why m is taken from -infinity to +infinity instead of just from 1 to +infinity.
(Does he say anything about m = 0?)

It would perhaps be more clear to use |6*m + 1| to indicate the sign of the prime is ignored. I guess he's trying to maintain a consistant clarity throughout.
:maybeso:

[QUOTE=Maybeso;90903]If I understand that part of troels "terminology" correctly (which I may not), numbers of the form (6*(+m) - 1) are constructed as -(6*(-m) + 1).
...
It would perhaps be more clear to use |6*m + 1| to indicate the sign of the prime is ignored. I guess he's trying to maintain a consistant clarity throughout.[/QUOTE]
I am afraid that Mr Munckner more than once denies this and this is why RDE asked Mr Munckner to respond in person. I am afraid that we will get another unclear and sidestepping answer.

I can not say that I see any constant "clarity" in his writing, alas :-(

Troels terminology.
 

[QUOTE=Maybeso;90903]Rde:

If I understand that part of troels "terminology" correctly (which I may not), numbers of the form (6*(+m) - 1) are constructed as -(6*(-m) + 1). Like this:
5 -> 6*(-1) + 1 = -5
11 -> 6*(-2) + 1 = -11
17 -> 6*(-3) + 1 = -17
23 -> 6*(-4) + 1 = -23

This is what he means by using +1 as the centre for primes and prime products. If he chose -1, then 5 = 6*1 - 1, and 7 = 6*(-1) - 1. That is why m is taken from -infinity to +infinity instead of just from 1 to +infinity.
(Does he say anything about m = 0?)

It would perhaps be more clear to use |6*m + 1| to indicate the sign of the prime is ignored. I guess he's trying to maintain a consistant clarity throughout.
:maybeso:[/QUOTE]

:surprised

Well maybeso, of all the irrelevant posts and criticisms on this subject, I

think you have taken the trouble to understand troels mathematics and lead us all somehwere.

You have hit the nail on the head this time. Congratulations!

I reiterate troels definition from my post #76

[Quote=troels]
I reiterate Troels definitions.

2) The Never Primes: These comprise all even numbers AND all odd numbers divisible by 3
On the number line NP are located symmetrically around 0 and so may be called 0-centrred integers.
NP constitute 2/3 of all numbers including two real primes No.s 2 and 3.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers.QUOTE]
Mally

I note that he is taking three centres for his 'zero'.

The zero centred integers and the +- 1 centred integers.

Zero is very much there but is used for 'Never primes' ( see point 2 of his )

For 'possible primes' +- 1 are used as starting points thus

For positive primes like 5 , 7 , 11 etc. use is made of -1 as the centre using

the formula 6M - 1

For negative primes he uses +1 as the the centre thus getting -5 , -7 , etc.

using the formula 6M + 1 [here m is negative]

Of all the replies, yours makes the most sense and we need you to clarify

further what seems to be anomalies. I am sure that with a little trouble to

study troels posts, and with consistency, these could be ironed out,

resulting in a beautiful theory on primes

So please stay tuned Maybeso and I'm passing the baton on to you but will

check troels theory every now and then.

Mally :coffee:

Troels Munkner's definition of a "possible prime", which includes all the primes he's able to recognise as so, is *not* |6m+1|. It's just 6m+1, which implies 5, 11, 17 and 23 aren't primes, or that he'll contradict himself:

[QUOTE=brunoparga;89361]I think the best way to find out what mr. Munkner means would perhaps be asking him plain, simple, easy-to-understand questions.

So, mr. Munkner, please classify both the statements below as either "true" or "false". Whatever your answers are, we've already understood the logic beneath them, so you don't need to spend time explaining why these statements are true or false.

1) The number 5 (positive five) is a prime number. True or false?
2) The number -5 (negative five) is a prime number. True or false?

Please do follow my guidelines strictly, as I personally have a very hard time understanding math which doesn't present itself to me according to them.

Thanks a lot,
Bruno[/QUOTE]

[QUOTE=troels munkner;89710]1 is false,
2 is true.

Y.s.
Troels Munkner[/QUOTE]

That is, he has more than once stated clearly that, according to his "theory", every integer must be either even, or divisible by 3, or a "possible prime" (=6m+1). Perhaps he should add another category, like this: even, 3-divisible, "possible prime" and the-ones-I-cannot-understand-why-people-keep-asking-me-about-them, and those would include the (real) primes == -1 (mod 6).

Bruno

[QUOTE=brunoparga;90946]Troels Munkner's definition of a "possible prime", which includes all the primes he's able to recognise as so, is *not* |6m+1|. It's just 6m+1, which implies 5, 11, 17 and 23 aren't primes, or that he'll contradict himself:





That is, he has more than once stated clearly that, according to his "theory", every integer must be either even, or divisible by 3, or a "possible prime" (=6m+1). Perhaps he should add another category, like this: even, 3-divisible, "possible prime" and the-ones-I-cannot-understand-why-people-keep-asking-me-about-them, and those would include the (real) primes == -1 (mod 6).

Bruno[/QUOTE]

He hasn't even presented a "theory". He has said nothing at all.
He has simply observed that the integers that are 1 mod 6 are closed
under multiplication and that they don't include 2 and 3. Big whoopeee.

[QUOTE=R.D. Silverman;90949]He hasn't even presented a "theory". He has said nothing at all.
He has simply observed that the integers that are 1 mod 6 are closed
under multiplication and that they don't include 2 and 3. Big whoopeee.[/QUOTE]

I totally agree with you. But up to now, nobady could convince troels about that, so I tried it with a very simple question (Im still waiting for an answer).
According to his definitions, if I understood them correctly, 5, 11, ... arent integers because they dont belong to one of his postulated groups. If they arent integers, they cant be used as values for m in his 6*m+1 formula, so I conclude that also 31 (6*5+1) and 67 (6*11+1) arent integers, what leads to a total chaos.

Mis-translation?
 

[QUOTE=brunoparga;90946]Troels Munkner's definition of a "possible prime", which includes all the primes he's able to recognise as so, is *not* |6m+1|. It's just 6m+1, which implies 5, 11, 17 and 23 aren't primes, or that he'll contradict himself:

That is, he has more than once stated clearly that, according to his "theory", every integer must be either even, or divisible by 3, or a "possible prime" (=6m+1). Perhaps he should add another category, like this: even, 3-divisible, "possible prime" and the-ones-I-cannot-understand-why-people-keep-asking-me-about-them, and those would include the (real) primes == -1 (mod 6).

Bruno[/QUOTE]
:smile:
Bruno, I think you caught him off guard on your simple questions. There is a language problem here.

Iwould say please give the man a chance and help him to develop his theory.
By criticism and sarcasm none of us can get anywhere.

But this is maths history all over again. Math'cians with original theories have received scathing attacks from other math'cians whose main purpose is to block the truth with ridicule and discouragement simply because they didnt or couldnt have the imagination to propound a theory themselves.
I am merely stating a historical fact and do not and will not engage in a controversy on this point.

If the moderator (and I appeal specially to Ernst ewmayer) can sieve out all non mathematical posts and comments in this thread I'm sure we could get a viable thread on prime numbers from not only Troels but other competent math'cians like Maybeso to unravel the skein of Troelsian mathematics.

What is the need of the hour is to have posters to look or scratch below the surface of the oxide and reveal the nugget below. Obviously throwing it back into the river cannot help much.

Bruno, for your benefit I quote below from Troels book.

3) Possible Primes (PP): These are all odd numbers which cannot be divided by 3.
PP are located symmetrically around +1 or – 1 depending on your choice.
These may be called 1-centred integers. [ For this refer to maybeso's and my posts]

Possible primes can be subdivided into real primes and prime products.

The possible primes constitute one third of all numbers [/QUOTE]

Please note that integers 5 , 7 , 11 etc. fall into this category.

Mally :coffee:

Thx Mally for your answer. This was my best guess too. But the language of troels is everything but clear, so confusion is programmed. But I still dont see any value in troels thoughts (I may not be the only one...)

97 posts and counting ...