number theory

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number, integer, Gauss, Legendre, Euler, Dedekind, Wiles, Weil, Grothendieck, Deligne, Faltings, Serre, prime
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Fermat's work in the 1800's

Can you help me with this...

Assume p is prime. Prove that p divides 2^p-2 .

Re: Fermat's work in the 1800's

Hi,
p = 2 is trivial. So assume p not equal 2, therefore gcd(2,p) = 1.
But you have 2[2^{p-1}-1]/p. So that you must to show 2^{p-1}-1 divides p. Do a left click on pahio's link, read, and you get it.
perucho

Re: Fermat's work in the 1800's

OOOPS!

2^{p-1}-1 divides p???? p divides 2^{p-1}-1 ! Sorry.

Sequence

Consider the sequence a[n]=4+7(n-1)=7n-3. In my text this sequence is represented as follows: a, a+d, a+2d, a+3d,... a+n(n-1)d,...

My comment is that the a+n(n-1)d terms represent some of the terms but not each and every successive term. Please feel free to comment. Thanks.

z

Re: Fermat's work in the 1800's

OK, I just multiplied Fermat's little theorem by a to get

a^p is congruent to a mod p which is the same form as

2^p is congruent to 2 mod p. So p divides a^p -p.

Correction

Sorry that's p divides a^p -a .

Must be typo.

Re: Sequence

What would be the typo? The equality 4 + 7(n - 1) = 7n - 3 checks out. To make sense of a, a + d, a + 2d, a + 3d, ... a + n(n - 1)d, ... plug in a = 4 and d = 7. Then the whole thing becomes just another way of writing a + (n - 1)d = 4 + 7(n - 1).

Re: Fermat's work in the 1800's

check out fermats little theorem