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For each $k$, is there something we can say about each positive integer being able to be written as a sum of $k$ positive squares? What is known for $k > 5$? What about if we only care about whether a prime can be written as a sum of $k$ positive squares for each given $k$?

During my investigation, I found a lot of results.

For $k=2$, a simple test on the prime factors and exponentes can be made, which can be seen in the "FORMULA" section of this OEIS page.

For $k=3$, it seems that no definitive rule has been found, but there are some interesting facts in this OEIS page

For $k=4$, I have found very few information in the OEIS page

For $k=5$, even less information is found in the OEIS page

I have also found this interesting OEIS page about sums of squares.

So what about $k > 5$? Is there anything interesting about the sum of $k$ positive squares? And what if we restrict ourselves to primes?

Arthur Queiroz Moura
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    But the "interesting" page says: "Please do not rely on any information it contains." For six positive squares and more see for example this post. – Dietrich Burde May 01 '23 at 19:15
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    better information for $k=4,5$ at the "complement" pages https://oeis.org/A000534 ....

    https://oeis.org/A047701 ...... general facts in Halter-Koch
    http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf

     – Will Jagy May 01 '23 at 19:38
  • I have the fifth edition of Niven, Zuckerman, Montgomery. The simple facts about four or five nonzero squares are on pages 317-319, especially Corollary 6.27. More detail about four is in Conway, The Sensual Quadratic Form, especially the Theorem on page 140 – Will Jagy May 01 '23 at 19:48

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