Conjecture: Any sufficiently big sum of three squares can be written as a square sum of three different natural numbers greater than zero

conjecturesreference-requestsums-of-squares

A natural number can be written as a sum of three squares if and only if it's not of the form $4^m(8n+7)$ for natural numbers $m,n$.

I'm curious about such numbers that can be written as the square sum of three different natural numbers $>0$. My conjecture is:

Any sufficiently big sum of three squares can be written as a square
sum of three different natural numbers greater than zero.

That is, there is a largest number $a^2+b^2+c^2$ which can not be written so that $0<a<b<c$. I would like to see a proof or a counter-proof or a related reference request.

See also Primes of the form $a^2+b^2+c^2$, $0<a<b<c$

Best Answer

OEIS/A004432 gives the following conjecture by Jeffrey Shallit:

A number is a sum of 3 squares, but not a sum of 3 distinct nonzero squares, if and only if it is of the form $4^js$, where $j \ge 0$ and $s \in \{1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 25, 27, 33, 34, 37, 43, 51, 57, 58, 67, 73, 82, 85, 97, 99, 102, 123, 130, 163, 177, 187, 193, 267, 627, 697\}$.

In particular, $4^j\cdot 3$ is an unbounded sequence of numbers that are not a sum of 3 distinct squares.

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