A (new) prime theorem.

[R.D. Silverman]

Originally Posted by troels munkner

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.

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?

[mfgoode]

Originally Posted by R.D. Silverman

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?

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

[Xyzzy]

Originally Posted by mfgoode

...I know I am throwing a straw to a drowning man...

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



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

[mfgoode]

Originally Posted by Xyzzy

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



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

It means I am not of much help but I offer hope.

Mally

[ewmayer]

Originally Posted by Xyzzy

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

Uh oh - looks like someone's been spending too much time in the Factoring forum...

[retina]

Originally Posted by Xyzzy

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

Could this be a forum bug? Or perhaps malware on Xyzzy's PC?

[Uncwilly]

Originally Posted by Uncwilly

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.

Correction:
B and D in base 16 would be fundamentals, but 11 and 13 would not be in base 10.

[ewmayer]

Originally Posted by retina

Could this be a forum bug? Or perhaps malware on Xyzzy's PC?

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.

[fetofs]

Originally Posted by xilman

This statement is both correct and trivial. It also defines the term "possible prime" to mean integers of the form . 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

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 .

[mfgoode]

Originally Posted by fetofs

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 .

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

[Terence Schraut]

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.