Famous Theorems of Mathematics/Number Theory/Prime Numbers

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This page will contain proofs relating to prime numbers. Because the definitions are quite similar, proofs relating to irreducible numbers will also go on this page.

Definition of Prime

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A prime number p>1 is one whose only positive divisors are 1 and p.

Basic results

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Theorem: is prime and implies that or .

Proof: Let's assume that is prime and , and that . We must show that .

Let's consider . Because is prime, this can equal or . Since we know that .

By the gcd-identity, for some .

When this is multiplied by we arrive at .

Because and we know that , and that , as desired.