Numbers of the form 24*k+1 appear to be perfect squares quite often. For instance, 25, 49, 121. Why?

Question

jonathan34 · Answer

bc idk lol

ganeshie8 · Answer

$$n^2 \equiv 0,1,4,9,12,16\pmod {24}$$

ganeshie8 · Answer

Out of those 6 possible residues, 

0 occurs 2 times,
`1 occurs 8 times,`
4 occurs 4 times,
9 occurs 4 times,
12 occurs 2 times,
16 occurs 4 times

ganeshie8 · Answer

Looks 1/3rd of the squares are of the form 24k+1

kainui · Answer

Oh interesting that it's so common like this, totally unrelated to what I was working on just happened to notice this thanks!

ganeshie8 · Answer

That still doesn't answer why 1 occurs that many times... This seems like a very broad q

kainui · Answer

This might be part of the reason why no odd perfect numbers have been found (supposing they exist) they would be required to have a prime number congruent to 1 or 17 mod 24, and if 1 mod 24 is usually a perfect square that sorta 'crowds out' the possibility of it being prime. Kinda vague sounding but just sorta playing around with it.

kainui · Answer

Maybe this has something to do with those quadratic residues I've heard about a long time back but never learned what they were, or maybe not?