Why did I just come across this?
by SomeonecoolLovesMaths, Feb 21, 2025, 11:02 AM
Denote the number of divisors of 
as 
, and write 
for the number of those divisors with 
.Let
![\[
n \;=\; 2^f \,p_1^{r_1}\,p_2^{r_2}\,\dots\,q_1^{s_1}\,q_2^{s_2}\,\dots
\]](http://latex.artofproblemsolving.com/a/4/7/a475044d6ab24aafd755983e1a032f1ba60e34aa.png)
where 
and 
.Let
be the number of ways can be represented as the sum of two squares. Then:![\[
r_2(n) \;=\; 0 \quad \text{if any of the exponents } s_j \text{ are odd.}
\]](http://latex.artofproblemsolving.com/6/9/e/69ed7d6f34edad1d7c6c43db34d5f01fecac26e7.png)
![\[
\text{If all } s_j \text{ are even, then }
r_2(n) \;=\; 4\bigl(d_1(n)\;-\;d_3(n)\bigr).
\]](http://latex.artofproblemsolving.com/e/2/b/e2b3062dd285bb074aed50da6742384379a34424.png)
Every positive integer can be expressed as the sum of four integer squares. That is, for every 
, there exist integers 
, 
, 
, and 
such that![\[
n = a^2 + b^2 + c^2 + d^2.
\]](http://latex.artofproblemsolving.com/c/8/3/c83c79a16be18252904181a74d836d79383afaa5.png)
This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, Feb 21, 2025, 12:17 PM
March 2025
February 2025