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:

- for every

*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**.

**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:

2

**n**+ 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^{2}**n**+ 1 -**n**^{2}= 2

**n**+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**.