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 .

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Tags: Math, Monday Math, Number Theory

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