Some arithmetic...

science_man_88 · 2014-07-14, 23:06

just asking things:

I'm guessing it's known that $(2^{p-1}\eq 1 \text { mod p^2}) == (2^{2n-1}-1\eq 2n^2+2n \text { mod 4n^2+4n+1})$ where 2n+1 =p; if so has anything useful come out of it ?

Zeta-Flux · 2014-07-14, 23:30

Better than that, the Wieferich condition is equivalent to $2^{(p-1)/2} \equiv \pm 1\pmod{p^2}$ . This makes it slightly easier to test for the condition.

The only place I know of where the Wieferich condition is really useful is in the first case of Fermat's last theorem and in the solution to Catalan's conjecture.

science_man_88 · 2014-07-14, 23:35

Quote:

Originally Posted by Zeta-Flux

Better than that, the Wieferich condition is equivalent to $2^{(p-1)/2} \equiv \pm 1\pmod{p^2}$ . This makes it slightly easier to test for the condition.

The only place I know of where the Wieferich condition is really useful is in the first case of Fermat's last theorem and in the solution to Catalan's conjecture.

Okay, Thanks for that, I just thought it might be useful for trying to pin down what n are possible to create such p instead of trying any p ( which as far as I know, and yes I don't know much if any, is how it works).

science_man_88 · 2014-07-15, 01:23

Quote:

Originally Posted by Zeta-Flux

$2^{(p-1)/2} \equiv \pm 1\pmod{p^2}$ .

Sorry for quoting this twice. One thing that just came to me is that this is equivalent of saying $2^n \eq \pm 1\pmod{4n^2+4n+1}$ and this equals $2^n-1 \eq 0 \pmod{4n^2+4n+1}$ or $2^n+1 \eq 0 \pmod{4n^2+4n+1}$ which when you consider that if 2m+1 divides 2k+1, $k \eq m \pmod{2m+1}$ we can bring this down to $2^{n-1}-1 \eq 2n^2+2n \pmod{4n^2+4n+1}$ or $2^{n-1} \eq 2n^2+2n\pmod{4n^2+4n+1}$ I'm I getting better or just making it worse ?

Zeta-Flux · 2014-07-15, 03:55

I certainly don't want to discourage you from pursuing these ideas, as not much progress has been made in understanding Wieferich primes! I would recommend reading the original paper of Wieferich, or possibly those that followed (I seem to remember one by Mirimanoff (sp?)), which introduced how these primes play a role in Fermat's last theorem. You may find a connection between what you are seeing and that problem.

R.D. Silverman · 2014-07-15, 16:26

Quote:

Originally Posted by science_man_88

just asking things:

I'm guessing it's known that $(2^{p-1}\eq 1 \text { mod p^2}) == (2^{2n-1}-1\eq 2n^2+2n \text { mod 4n^2+4n+1})$ where 2n+1 =p; if so has anything useful come out of it ?

Would a moderator please move this (and succeeding) gibberish to the crank
math sub-forum?

science_man_88 · 2014-07-15, 19:03

Quote:

Originally Posted by R.D. Silverman

Would a moderator please move this (and succeeding) gibberish to the crank
math sub-forum?

can you explain to me why you think it's gibberish ?

science_man_88 · 2014-07-15, 20:41

Quote:

Originally Posted by Zeta-Flux

I certainly don't want to discourage you from pursuing these ideas, as not much progress has been made in understanding Wieferich primes! I would recommend reading the original paper of Wieferich, or possibly those that followed (I seem to remember one by Mirimanoff (sp?)), which introduced how these primes play a role in Fermat's last theorem. You may find a connection between what you are seeing and that problem.

I found the title but not the paper itself the real problem after that is to translate it to English if I can't find an English copy. Do you know where I can find a free copy of Zum letzten Fermat'schen Theorem ?

science_man_88 · 2014-07-15, 23:52

Quote:

Originally Posted by science_man_88

I found the title but not the paper itself the real problem after that is to translate it to English if I can't find an English copy. Do you know where I can find a free copy of Zum letzten Fermat'schen Theorem ?

nevermind I found a free preview of some of it and read a bit on wikipedia. I did fail so far to find specific n candidates that work, I'll give RDS that. edit: the part about $m\leq 113$ sounds interesting for me since I see a way to generalize my first post to use those m. but I don't think it will help.

science_man_88 · 2014-07-28, 22:04

Quote:

Originally Posted by R.D. Silverman

Would a moderator please move this (and succeeding) gibberish to the crank
math sub-forum?

I decided I better show you how I got that:

the fact that if 2n+1|2k+1 (| meaning divides), $k \eq n \pmod{2n+1}$ can be shown by showing 2n+1 divides when k=n and that 2n+1 divides one in every 2n+1 odd numbers.
based on 2n+1=p we get: $2^{p-1}\eq 1 \pmod {p^2} == 2^{2n+1-1}\eq 1 \pmod {{2n+1}^2} == 2^{2n+1-1}\eq 1 \pmod{4n^2+4n+1} == 2^{2n}\eq 1 \pmod{4n^2+4n+1} == 2^{2n}-1\eq 0 \pmod{4n^2+4n+1$
based on these other two we can show that if $2^{2n}-1\eq 0 \pmod{4n^2+4n+1}$ then $2^{2n-1}-1 \eq 2n^2+2n \pmod{4n^2+4n+1}$

therefore the original congruence plus the basics of division can show that what I said is true. edit: this last result can be rewritten as $2^{p-2}-1 \eq {\frac{p-1}{2}(p+1)} \pmod {p^2}$

science_man_88 · 2014-07-30, 19:38

I'd like a double check on this but based on my last congruence I believe I've found a way to prove p can not be the second prime in a twin prime pair. (Q,p)

I'd love your feedback since I feel like I'm talking to myself.

2014-07-14, 23:06	#1
science_man_88 "Forget I exist" Jul 2009 Dumbassville 20300₈ Posts	Some arithmetic... just asking things: I'm guessing it's known that $(2^{p-1}\eq 1 \text { mod p^2}) == (2^{2n-1}-1\eq 2n^2+2n \text { mod 4n^2+4n+1})$ where 2n+1 =p; if so has anything useful come out of it ? Last fiddled with by science_man_88 on 2014-07-14 at 23:13 Reason: took a +1 away after realizing the error.

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2014-07-14, 23:30	#2
Zeta-Flux May 2003 7×13×17 Posts	Better than that, the Wieferich condition is equivalent to $2^{(p-1)/2} \equiv \pm 1\pmod{p^2}$ . This makes it slightly easier to test for the condition. The only place I know of where the Wieferich condition is really useful is in the first case of Fermat's last theorem and in the solution to Catalan's conjecture.

2014-07-15, 03:55	#5
Zeta-Flux May 2003 7×13×17 Posts	I certainly don't want to discourage you from pursuing these ideas, as not much progress has been made in understanding Wieferich primes! I would recommend reading the original paper of Wieferich, or possibly those that followed (I seem to remember one by Mirimanoff (sp?)), which introduced how these primes play a role in Fermat's last theorem. You may find a connection between what you are seeing and that problem.