A. Let =2 +1 (2 (Power 2(power n))) Plus 1. Prove that P is a prime Dividing , then the smallest m such that P (2 -1) is m = 2 (hint use the
Division Algorithm and Binomial Theorem)
Please see attached.© BrainMass Inc. brainmass.com March 4, 2021, 6:24 pm ad1c9bdddf
Please see the attachment.
First, I claim that if . Since , then we have
Second, I show that is the smallest integer which satisfies . Suppose the smallest is some , such that . Then we know , for any . So we have , (mod ). is the smallest one, then we must have . This implies that for some . Since , then (mod ). Since ...
There are a variety of proofs in this solution regarding modular arithmetic.