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Statement

Definitions (pp. 7--8). A primitive residue class $n\equiv r\pmod q$ (with $r$ coprime to $q$) is solvable by polynomials if there are polynomials $P_1(n),P_2(n),P_3(n)$, positive-integer-valued for all large $n$ in the class (so with rational coefficients), with $4/n=1/P_1(n)+1/P_2(n)+1/P_3(n)$ for all such $n$. It is Type I solvable if this can be done with exactly one of the $P_i$ having no constant term, and Type II solvable if exactly two have none; the paper shows every solvable primitive class is one or the other (p. 8).

Proposition 1.9 (Solvable congruences), p. 8. The print names the class "$q\bmod r$", although the definitions just before it write $r\bmod q$. Let the primitive residue class be given.

1. If it is Type I solvable by polynomials, then all sufficiently large primes in it lie in one of finitely many residue classes taken from the following families:

   • $n\equiv-f\pmod{4ad}$, where $a,d,f\in\mathbb N$ and $f\mid4a^2d+1$;
   • $n\equiv-f\pmod{4ac}$ and $n\equiv-c/a\pmod f$, where $a,c,f\in\mathbb N$ and $(4ac,f)=1$;
   • $n\equiv-f\pmod{4cd}$ and $n^2\equiv-4c^2d\pmod f$, where $c,d,f\in\mathbb N$ and $(4cd,f)=1$;
   • $n\equiv-1/e\pmod{4ab}$, where $a,b,e\in\mathbb N$, $e\mid a+b$ and $(e,4ab)=1$.

   Conversely, every residue class in one of these four families is solvable by polynomials.

2. If it is Type II solvable by polynomials, then all sufficiently large primes in it lie in one of finitely many residue classes taken from the following families:

   • $-e\bmod 4ab$, where $a,b,e\in\mathbb N$, $e\mid a+b$ and $(e,4ab)=1$;
   • $-4a^2d\bmod f$, where $a,d,f\in\mathbb N$ and $4ad\mid f+1$;
   • $-4a^2d-e\bmod 4ade$, where $a,d,e\in\mathbb N$ and $(4ad,e)=1$.

   Conversely, every residue class in one of these three families is solvable by polynomials.

The paper attributes most of these families to earlier work and says one condition in the list appears to be new (p. 9); it says the proposition "essentially classifies all solvable primitive congruences" (p. 8).

Source. Elsholtz and Tao, arXiv:1107.1010v6, p. 8 (the definitions on pp. 7--8); read on the page images. Proved in Section 10 (pp. 37--40). The arXiv comment on v6 says its statement and proof were corrected (the comment calls it Theorem 1.9). Published as J. Aust. Math. Soc. 94 (2013), no. 1, 50--105, DOI 10.1017/S1446788712000468; the published version was not compared.

Read depth. Claims checked: the definitions, the seven families with their conditions and the two converse clauses were read clause by clause; the proof was not read.

Proof pointer

Section 10 first checks that each family is solvable by polynomials, then shows that the large primes of a Type I or Type II solvable class fall into the listed families.

Dependencies

The proof in Section 10; not examined here.

Bears on

• Problem 242: the proposition describes which residue classes can be settled by polynomial identities, the method behind Mordell's and similar partial results. The paper notes (p. 8) that every primitive class modulo 840 is solvable by polynomials unless $r$ is a perfect square, and that a class of perfect squares is not (citing earlier work, and as a consequence of Proposition 1.6); the paper reads that vanishing at odd squares (p. 6) as ruling out strategies such as a finite set of covering congruences. The proposition proves no new case of the conjecture.