# Modular proofs

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.

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#### Solution Preview

Please see the attachment.

a. Proof:

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 ...

#### Solution Summary

There are a variety of proofs in this solution regarding modular arithmetic.

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