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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.

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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
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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
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@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. –  Andres Caicedo Mar 11 '12 at 18:58
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original 1959 ams.org/journals/proc/1959-010-03/home.html –  Will Jagy Mar 11 '12 at 19:57
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original 1933 Gordon Pall dm.unito.it/~cerruti/ntlab2007/squares-pall.pdf –  Will Jagy Mar 11 '12 at 20:09
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1 Answer

up vote 7 down vote accepted

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.)

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Thanks! To complement: The paper linked to in the answer is "Sums of Squares and Orthogonal Integral Vectors" by Lee M. Goswick, Emil W. Kiss, Gábor Moussong, and Nándor Simányi. Journal of Number Theory, Volume 132, Issue 1, January 2012, Pages 37-53. There they discuss (among other topics) the following notion: A positive integer is twin-complete if and only if its square-free part is twin-complete, where a positive square-free integer is twin-complete if and only if it can be written as a sum of two squares, but not as a sum of three positive squares. ... –  Andres Caicedo Mar 11 '12 at 22:30
    
... 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). ... –  Andres Caicedo Mar 11 '12 at 22:34
    
... The conjecture is reviewed, with a good number of references, in the last section of the paper (Section 6). This clearly answers my question. Once again, many thanks. –  Andres Caicedo Mar 11 '12 at 23:46
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