Sums of three non-zero squares

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.

• Is this assuming "just" Riemann or Extended Riemann (i.e. for quadratic Dirichlet characters)? I don't know the Grosswald paper, but it seems from the list that the question comes down to the existence of a large idoneal number under the further condition that it be a sum of two squares, and there's likely no way to exploit this additional assumption, so it probably remains open. – Noam D. Elkies Mar 11 '12 at 17:08
• 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. – Barry Cipra Mar 11 '12 at 17:57
• @Noam: Looking at Grosswald's paper from 1963, what he uses is a weak version of Extended Riemann. The reference Barry Cipra mentioned and some of the papers they cite indicate the issue is exactly what you indicate. – Andrés E. Caicedo Mar 11 '12 at 18:58
• original 1959 ams.org/journals/proc/1959-010-03/home.html – Will Jagy Mar 11 '12 at 19:57
• original 1933 Gordon Pall dm.unito.it/~cerruti/ntlab2007/squares-pall.pdf – Will Jagy Mar 11 '12 at 20:09

• ... They restate the question above in this language (their Conjecture 1.9). Then they comment that the elements of the set $T$ "form a subset of Euler’s numeri idonei, and therefore, at most one number can be absent from the list above. If such an integer does exist, it must exceed $2\cdot 10^{11}$ (P. J. Weinberger, "Exponents of the class groups of complex quadratic ﬁelds", Acta Arith. 22 (1973), 117–124), and if it is even, the Generalized Riemann Hypothesis is false (J. Borwein, K. K. S. Choi, "On the representations of $xy + yz + zx$", Exp. Math. 9 (2000), 153–158). ... – Andrés E. Caicedo Mar 11 '12 at 22:34