mersenneforum.org - View Single Post - Quadratic residue counts related to four squares representation counts

Till · 2020-09-20, 12:23

Hi all,
today I found something curious...

Let X(m) be the number of distinct quadratic residues mod m. (A023105 for m=2^n)
Let Y(m) be the number of n < m that can be expressed as a sum of 4 squares, but not by a sum of less than four squares. (A004015)

Then it appears that X(2^n) == Y(2^n) + 2 for all n.

A simple Java test program can be found here:
https://github.com/TilmanNeumann/jav...f4Squares.java

2020-09-20, 12:23	#1
Till "Tilman Neumann" Jan 2016 Germany 494₁₀ Posts	Quadratic residue counts related to four squares representation counts Hi all, today I found something curious... Let X(m) be the number of distinct quadratic residues mod m. (A023105 for m=2^n) Let Y(m) be the number of n < m that can be expressed as a sum of 4 squares, but not by a sum of less than four squares. (A004015) Then it appears that X(2^n) == Y(2^n) + 2 for all n. A simple Java test program can be found here: https://github.com/TilmanNeumann/jav...f4Squares.java Last fiddled with by Till on 2020-09-20 at 12:33 Reason: A023105