Grant MacEwan University puts on noontime math lectures every second Friday. Sometimes I skip Economics, and Tim and I go. At the end of every lecture, cards are distributed with a math problem to be solved for the next lecture, and everyone who hands in a solution during the next two weeks wins a box of doughnuts. Yesterday I got my doughnuts [horrifyingly, I was the first girl to submit a solution all year] for the first formal proof I've ever written. I'm inordinately proud of it, so here it is:
Show that for every prime number p, where p does not equal 2, there exist integers a and b so that a 2 - b 2= p.
- for every p such that p > 2 and p is prime, p is not divisible by 2 [by the definition of a prime number], so p is odd
- every odd integer can be expressed as 2 times an integer plus 1; thus:
2n + 1 or n + (n+1)
- for every set of two adjacent integers n, (n+1), where a = (n+1) and b = n,
a 2 - b 2
= n 2 +2n + 1 - n 2
= 2n +1
- every odd integer can be expressed in the form a 2- b 2 = p. Since all primes > 2 are odd, for every prime p, where p does not equal 2, there exist integers a and b so that a 2 - b 2 = p.