Let p be an odd prime of the form p = 4k + 3 a) Prove that if q = 2p + 1 is prime, then q divides the Mersenne number . b) Use part (a) to prove that , , and are composite numbers.
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(b) should be easy enough since I know that everything from (a) must hold and I have specific examples. I don't really get what I'm supposed to use to prove (a). What does q being prime have to do with it dividing the Mersenne number?
a. If p=4k+3 then q=8k+7 q|M_n , in other words is: 2^{(q-1)/2}=2^p \equiv 1(mod q) It is actually the same to say that Legendre's symbol (2/q)=1. {(2/q)=1, when q=1(mod 8) or q=7(mod 8)}
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