Sums of products of powers of given prime numbers

Robert Tijdeman and Lian Xiang Wang

Vol. 132 (1988), No. 1, 177–193

DOI: 10.2140/pjm.1988.132.177

Abstract

We give the complete solutions of the equations 2^x3^y + 1 = 2^z + 3^w, 2^x3^y + 2^z = 3^w + 1 and 2^x3^y + 3^w = 2^z + 1 in integers x, y, z, w. We use this to prove that every large rational number has at most four representations of the form 2^α3^β + 2^γ + 3^δ. Finally we prove that, for given integer n and prime numbers p₁,…,p_t, every rational number m has at most C representations of the form ∑ _i=1ⁿp₁^k_i1⋯p_t^k_it where k_i1,…,k_it are integers.

Mathematical Subject Classification 2000

Primary: 11D61

Secondary: 11D85

Milestones

Received: 24 October 1986

Revised: 18 March 1987

Published: 1 March 1988