I've been able to prove that the set {8n+7} has infinite primes by manipulating Fermat's Theorem, but I'm trying to reprove it using quadratic residue and Legendre Polynomials.
I've been able to show that for p=8n+7, (2/p)=1 and (-1,p)=-1
So it follows that (-2/p)=-1. And that (-2/p)=1 iff...
Just a couple questions that I'd appreciate any help on.
1. if [(2^d) - 1] is prime, prove that d is prime as well.
2. Prove that (p-1)C(k) is congruent to (-1)^k mod p.
I've started them both but ended up getting stuck.
Any ideas?
Thanks
(a^m^2 + 1) | (a^n^2 - 1) ?
I'm sure there is a quick trick I'm missing somewhere... but anyone have any ideas on how to prove:
(a^m^2 + 1) | (a^n^2 - 1) , for n > m.
[Show [a^(n^2) -1] is divisible by [a^(m^2) +1]
Thanks a lot. . .
(I've tried letting k=n-m, and other stuff like...
If you can help, that would be great.
Let R be a commutative ring, and A,B,M be R-modules. Prove:
a) HomR(A x B, M) is isomorphic to HomR(A, M) x HomR(B, M)
b) HomR(M, A x B) is isomorphic to HomR(M, A) x HomR(M, B)