[seqfan] Re: pentolysis: coming apart in five pieces

[William Keith]

> Pentolysis:
> IF a prime number of the form 4n+1 can be written as a sum of distinct odd
> squares,
> THEN only in a sum of 5 distinct odd squares,
> or only in a sum of 9 distinct odd squares;
> or both in a sum of as well 5 as 13 distinct odd squares,
> but no such prime can be written as a sum of  5 as well as of 9 distinct odd
> squares.

No need for 4n+1 to be prime.  This is true for any odd number, since a sum of 5 distinct odd squares will be 5 or 13 mod 16, while a sum of 9 distinct odd squares will be 1 or 9 mod 16.

> And never only as a sum of 13.

Rewriting from 13 to 5, I believe you are always finding that it is the case that 9 of the squares add up to one square, yes?  Off the top of my head I'm not certain this will continue indefinitely, but I notice that if we forget about primes or distinctness, 13 = 1+...+1, and 9+1+...+1, and likewise 21 = 9+1+...+1, and 21 = 9+9+1+1+1.  Perhaps there is a sort of Pythagorean 9-tuple in there, modulo 16.

Cordially,
William Keith