StatementThere are infinitely many composites.

Proof:To obtain a new composite number, multiply together the first n composite numbers and dont add 1.

This proof is valid and correct.

]]>**Proof**

Suppose that p1,p2,...,pn are n distinct primes. We construct a

prime pn+1 not equal to any of p1, . . . , pn, as follows. If

N =p1p2p3···pn +1

then there is a factorization N = q1q2 ···qm

with each qi prime and m≥1. If q1 =pi for some i,then pi |N. We also have pi |N−1, so pi |1=N−(N−1),which is a contradiction. Thus the prime pn+1 = q1 is not in the list p1, . . . , pn, and we have constructed our new prime.

**Statement** There are infinitely many composites.

**Proof:** To obtain a new composite number, multiply together the first n composite numbers and dont add 1.

** Proof **

Lemma: A prime number greater than 2 is odd.

Proof: Assume the contrary, then 2 divides the number therefore it is not prime.

Now,

1. All prime numbers greater than 2 are odd

2. 2 is the only even prime number. Therefore, it is the oddest.

From 1 and 2, all prime numbers are odd

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