# Cryptography, congruences, and primes

Hi,

Can you help me with these questions?

Consider the set of all even integers 2Z=....If this factorization into primes can be accomplished, is it unique? (see attached)

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#### Solution Preview

Part I: Prime Numbers

(a) By the definition, all the primes in is , which is the set of the multiplications of 2 and an odd integer.

(b) Yes. We know, each even positive integer can be expressed as , where is an odd integer and . Since and , then we have

, the multiplication of 2's and .

So each even positive integer can be expressed as a product of these primes.

(c) This factorization is not unique.

Here is a counter example. Let , then we have

and

So has two ways to be expressed as the product of primes.

Part II: Congruence

Problem #7

Proof: If is a perfect square, then , then (mod 5). But we can go through all residues in modulo 5 and find that is not a quadratic residue. Therefore, (mod 5) has no solutions. This is a contradiction.

Hence can not be a perfect square.

Problem #46

in ...

#### Solution Summary

This provides examples of working with factorization into primes of even integers, proving an integer of a given form is not a perfect square, modular arithmetic, and RSA ciphers.