---

Claim. In Lattice points on circles, squares in arithmetic progressions and sumsets of squares (Additive Combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc., 2007, 241-262; arXiv v1, p. 4), Cilleruelo and Granville show that the Bombieri-Lang conjecture implies Solymosi's conjecture. That conjecture says that for some $d$, no affine cube $\{b_0+\sum_{i\in I}b_i: I\subseteq\{1,\dots,d\}\}$ with nonzero $b_i$ consists of distinct squares. In such a cube, each element $x^2$ of the subcube over $\{3,\dots,d\}$ makes $(x^2+b_1)(x^2+b_2)(x^2+b_1+b_2)$ a square. So at least $2^{d-2}$ integers $x$ make that product a square, and Bombieri-Lang, through Caporaso, Harris and Mazur, bounds that number uniformly for polynomials of degree five or six without repeated roots. The site reads this as a negative answer to the second question of Problem 782. Since a positive answer to the first question implies one to the second, the first would also be negative. The library card is Cilleruelo and Granville 2007, and the conjecture with this derivation is recorded on Conjecture 6.

The Bombieri-Lang conjecture is unproved. No acceptance evidence is on record: the volume is a proceedings volume with no evidence of refereeing, and the site labels the problem OPEN.