A (new) prime theorem.

[mfgoode]

Originally Posted by Jens K Andersen

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 here or here. And delete his duplicate post in Information & Answers.

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

[S485122]

Originally Posted by mfgoode

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

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.

[xilman]

Originally Posted by R.D. Silverman

Note that the integers that are 1 mod 6 form an algebra in which unique multiplication FAILS.

Minor typo? I think you meant unique factorization.


Paul

[ewmayer]

Originally Posted by mfgoode

Maybe Troels finds it difficult to describe his concepts and we must have patience and give him the benefit of the doubt.

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:

1) Outrageous-sounding claims;

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

3) Vague/Hard-to-Find references in place of clear and present argumentation: 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.

[ewmayer]

Originally Posted by xilman

Minor typo? I think you meant unique factorization.

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.

[troels munkner]

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

[Wacky]

Originally Posted by troels munkner

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).

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

[mfgoode]

Originally Posted by troels munkner

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

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

[Fusion_power]

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

[troels munkner]

Originally Posted by Fusion_power

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

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

[troels munkner]

Originally Posted by Fusion_power

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

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