It is a well-known result of Legendre that a positive integer is sum of three squares unless it is of the form $4^a(8b+7)$.
In
- Grosswald, E.; Calloway, A.; Calloway, J. The representation of integers by three positive squares. Proc. Amer. Math. Soc. 10 1959 451–455. (MR0104623 (21 #3376)),
it is shown that there is a finite set $T$ such that any positive integer is a sum of three non-zero squares unless $n$ is of the form $4^a(8b+7)$ or of the form $4^am$ where $m\in T$.
The set $T$ is essentially identified, see
- Grosswald, Emil. Representations of integers as sums of squares. Springer-Verlag, New York, 1985. xi+251 pp. ISBN: 0-387-96126-7 (MR0803155 (87g:11002)):
Either
$$ T=\{1,2,5,10,13,25,37,58,85,130\}, $$
or else the Riemann hypothesis fails, and $T$ consists of these 10 numbers, and at most another one, $k$, that must be larger than $5\cdot10^{10}$. The conjecture is that $|T|=10$, of course.
I could not find any updates on the question of whether the conjecture has been settled, and would appreciate any information or pointers to the relevant literature.
Best Answer
Googling on the title of the Grosswald paper produced a link to math.uab.edu/~simanyi/Goswick_et_al_final.pdf which (backing up the url to the ~simanyi) indicates it's a recent paper in JNT. (Re-posted from comments at the OP's suggestion.)