It's better I think to shift to math is fun. And I have so many math ideas where you can just play with numbers and see things for yourself.

Like counting quadratic residue pairs! I didn't even know what a quadratic residue pair was until recently, because of a discovery.

Quadratic residues are the squares of modular arithmetic, like with x

^{2}= r mod N, r is the quadratic residue.

Here are the quadratic residues for 29:

1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28

Some are explicitly squares, like 4, 9 and 16, while others are not as you flip over when you get past 29, so for instance 6

^{2}= 36 = 7 mod 29.

And some of the quadratic residues follow each other, like 4 and 5 and 5 and 6, and math people call those quadratic residue pairs and I figured out a way to count them. There was a way known before, but I found my own way, which I like better.

The end equations have to be the same, of course.

And if p is prime, and p mod 4 = 1, the count is (p-5)/4, which for 29 is 6.

And here are the pairs with participating numbers in bold. Note that 4,5 is a pair, as is 5,6, so they count as two.

1,

**4**,

**5**,

**6**,

**7**, 9, 13, 16, 20,

**22**,

**23**,

**24**,

**25**, 28

Go with what people can prove. If you like playing with numbers there are lots of things on this blog, where I explain and derive. I'm not a mathematician though, and am not trained in writing things in some peculiar format, so I have my own style. Proof is rigorous though.

And besides the numbers speak more eloquently. There can be such a thrill in watching numbers follow these beautiful mathematical rules. Math IS fun.

All of the above is the confident, act like I really know what I'm doing kind of thing, but at the end, yeah, it is about the numbers. If your math is right, the numbers behave by stated rules. And for some, the precision and beauty of numbers IS all that's really important when it comes to discussing useful mathematics.

James Harris