A (new) prime theorem. - Page 6

ewmayer · 2006-10-25, 16:26

In a similar nomenclatural vein, I propose that all integers divisible by 11 be named "Tufnel 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^p-1 is not prime. It's "one more divisible", isn't it?

Phil MjX · 2006-10-25, 21:15

Quote:

Originally Posted by ewmayer

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^p-1 is not prime. It's "one more divisible", isn't it?

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

Ethan Hansen · 2006-10-26, 16:18

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 that 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.

CRGreathouse · 2006-10-26, 16:58

Quote:

Originally Posted by ewmayer

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^p-1 is not prime. It's "one more divisible", isn't it?

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.

Terence Schraut · 2006-10-26, 18:51

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.

xilman · 2006-10-26, 18:55

Quote:

Originally Posted by Terence Schraut

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.

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.

ewmayer · 2006-10-26, 19:08

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 post above. Enjoy...

Phil MjX · 2006-10-27, 06:44

Quote:

Originally Posted by CRGreathouse

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.

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

troels munkner · 2006-10-27, 11:12

Quote:

Originally Posted by CRGreathouse

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.

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

S485122 · 2006-10-27, 12:00

Quote:

Originally Posted by troels munkner

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.

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?

Uncwilly · 2006-10-27, 13:16

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.

2006-10-25, 16:26	#56
ewmayer ∂²ω=0 Sep 2002 República de California 19×613 Posts	In a similar nomenclatural vein, I propose that all integers divisible by 11 be named "Tufnel 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^p-1 is not prime. It's "one more divisible", isn't it? Last fiddled with by ewmayer on 2006-10-25 at 21:29 Reason: Note to Phil: Trying to pronounce "MjX" in any language is a challenge, please forgive me

2006-10-26, 16:18	#58
Ethan Hansen Oct 2005 2³·5 Posts	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 that 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.

2006-10-26, 18:51	#60
Terence Schraut Oct 2006 7 Posts	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.

2006-10-26, 19:08	#62
ewmayer ∂²ω=0 Sep 2002 República de California 11647₁₀ Posts	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 post above. Enjoy... Last fiddled with by ewmayer on 2006-10-26 at 19:28

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2006-10-27, 13:16	#66
Uncwilly 6809 > 6502 """"""""""""""""""" Aug 2003 101×103 Posts 7·23·61 Posts	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.