A conjecture of Erdős, supersingular primes and short character sums. (English) Zbl 1440.11039

In the paper under review, the authors consider the Diophantine equation which involves products of consecutive terms of positive integers in arithmetic progression and perfect powers. For \( k \) sufficiently large positive integer, they show that the Diophantine equation \[ n(n+d)(n+2d)\cdots(n+(k-1)d) = y^{\ell}, \quad \gcd (n,d)=1,\tag{1} \] has at most finitely many solutions in positive integers \( n,d,y, \) and \( \ell \) with \( \ell\ge 2 \).
Consider the conjecture of Erdős, which states that “there is a constant \(k_0\) such that the Diophantine equation (1) has no solutions in positive integers \(n, d, k, y, \ell,\) with \(\ell \ge 2\) and \(k \ge k_0\)”. The main result in this paper is a somewhat weak form of this conjecture, which also deals with the negative solutions of the equation (1) stated as follows.
Theorem 1. There is an effectively computable absolute constant \( k_0 \) such that if \( k\ge k_0 \) is a positive integer, then any solution in integers to the Diophantine equation (1) with prime exponent \( \ell \) satisfies either \( y=0 \) or \( d=0 \) or \( \ell \le \exp(10^{k}) \).
To prove Theorem 1, the authors rely upon Galois representations associated to certain Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial. In particular, the proof of Theorem 1 makes essential use of a wide array of tools from arithmetic geometry, analytic number theory and additive combinatorics.


11D61 Exponential Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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