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In this question : Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $?

I asked for the classification of the natural numbers being the sum of three positive perfect squares.

I slightly reformulate the question :

Is every positive integer $n$ of the form $4k+1$ or $4k+2$ ($k$ integer) greater than $130$ expressible as $a^2+b^2+c^2$ with positive integers $a,b,c$ ?

This seems to be the case and would already completely classify the natural numbers with the desired property. If $n$ is not a square, the following statement would be sufficient :

If $n>9634$ of the form $4k+1$ or $4k+2$ is not a square , then there is a positive integer $a$, such that $n-a^2$ is a prime number of the form $4k+1$. I found no counterexample upto $n=10^8$ and for sufficient large $n$ , there should be at least a heuristical argument making the existence of such an $a$ likely.

But what about the squares (which must then be of the form $4k+1$) ? Is every odd perfect square greater than $25$ the sum of three positive squares ? I found no counterexample upto $(10^4)^2=10^8$. Any ideas how to find at least a good heuristic that this statement is likely to be true ?

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  • $\begingroup$ You may be interested in OEIS sequence A000408, information, references, and links in it. $\endgroup$ – Somos Aug 2 '18 at 14:16

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