Squares that cannot be shown as sum of squares

Question

How many $n \in \mathbb{N}$ are there so that there exists no such $M \in \mathbb{N}$ so that $n^2 =\sum_{i=0}^{M}{a_i^2}$ for distinct $a_i \in \mathbb{N}$?

Source: http://mishabucko.wordpress.com

Answer 1

Halter-Koch [HK] showed that the largest integer not expressible as a sum of $5$ distinct non-zero squares with greater common divisor $1$ is $N^{*}(5)=245$. You can check the fifteen distinct squares $\le 15^2=225$ by hand. I find that

• $5^2 = 3^2 + 4^2$
• $7^2 = 2^2 + 3^2 + 6^2$
• $9^2 = 2^2 + 4^2 + 5^2 + 6^2$
• $10^2 = 6^2 + 8^2$
• $11^2 = 2^2 + 6^2 + 9^2$
• $13^2 = 3^2 + 4^2 + 12^2$
• $14^2 = 4^2 + 6^2 + 12^2$
• $15^2 = 9^2 + 12^2.$

So unless I'm missing one, the only squares not expressible as sums of distinct squares are these seven: $\{1^2,2^2,3^2,4^2,6^2,8^2,12^2\}$.

[HK] F. Halter-Koch, Darstellung naturlicher Zahlen als Summe von Quadraten, Acta Arith. 42 (1982), 11–20.