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Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero?


By the three square theorem of Legendre a natural number can be written as a square sum of three natural numbers if and only if it isn't of the form $4^i(8j+7)$. where $i,j$ are natural numbers.

Due to an answer to Conjecture: Any sufficiently big sum of three squares can be written as a square sum of three different natural numbers greater than zero there is a 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}

So, if the conjecture by Shallit is true, then all primes not of the form $8m+7$ and not belonging to $\{2, 3, 5, 11, 13, 17, 19, 37, 43, 67, 73, 97, 163, 187, 193\}$ can be written as a square sum of three different non zero natural numbers.

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closed as off-topic by user21820, Shailesh, nmasanta, Lee David Chung Lin, The Count Sep 3 at 0:02

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  • $\begingroup$ What is your motivation to ask this question? Does it appear somewhere else? $\endgroup$ – Jungle Boy Aug 6 at 13:10
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    $\begingroup$ @DavoodKhajehpour - Curiosity. $\endgroup$ – Lehs Aug 6 at 20:37
  • $\begingroup$ Dear Lehs, Do you know the relation between this kind of numbers [numbers represented as the sum of 3 squares] and physics? I heard these numbers are related to the energy levels of atomic particles, but I am not sure. This comment tells something related. $\endgroup$ – Jungle Boy Aug 7 at 8:01
  • $\begingroup$ Also, Long ago I asked 2 questions regarding this relation here & here. $\endgroup$ – Jungle Boy Aug 7 at 8:02
  • $\begingroup$ @DavoodKhajehpour - No, I didn't know that. $\endgroup$ – Lehs Aug 8 at 8:39
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Nothing seems be known. See OEIS/A125516. Contrast with OEIS/A085317.

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