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if 2 power p minus 1 is prime prove that p is prime.Is converse true?

if 2 power p minus 1 is prime prove that p is prime.Is converse true?

Grade:12

1 Answers

Rohit Suri
37 Points
8 years ago

Proof.
Let r and s be positive integers, then the polynomial xrs-1 is xs-1 times xs(r-1) + xs(r-2) + ... + xs + 1.  So if n is composite (say r.swith 1<s<n), then 2n-1 is also composite (because it is divisible by 2s-1).
Notice that we can say more: suppose n>1. Since x-1 divides xn-1, for the latter to be prime the former must be one. This gives the following.
Corollary.
Let a and n be integers greater than one.  If an-1 is prime, then a is 2 and n is prime.

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