Erdős Problem #1148

Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\max(x^2,y^2,z^2)\leq n$?

number theory

Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.

The largest integer known which cannot be written this way is $6563$. [Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$.

View the LaTeX source

This page was last edited 26 January 2026.

External data from the database - you can help update this
Formalised statement? Yes
Related OEIS sequences: A390380 A393168

6 comments on this problem

Likes this problem	old-bielefelder, ebarschkis
Interested in collaborating	None
Currently working on this problem	old-bielefelder
This problem looks difficult	None
This problem looks tractable	None
The results on this problem could be formalisable	None
I am working on formalising the results on this problem	None

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #1148, https://www.erdosproblems.com/1148, accessed 2026-03-01