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.



Sums of three non-zero squares
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.)


