## Monday Math 32

Today we have a bit of number theory. Let n be an integer greater than one. Then if  is prime, it can be expressed in the form , for some integer k.

Proof: we have that if  is prime,  for some integer k. The latter equation is the same as , which has a solution for integer k if and only if n is even. Now, we see that if n is odd, then  is odd, and so  is even. Since n>1, , and thus we see if n is odd,  cannot be prime. Thus, if  is prime, n is even, and so  is divisible by four, and thus  can be expressed in the form .