Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture

2025/08/25 by Le, Maohua, Miyazaki, Takafumi
#11D61 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.17601

Abstract

We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation n^x=(n+1)^y+(n+2)^z+(n+3)^w in positive integers n,x,y,z and w are given by (n,x,y,z,w)=(2,5,1,1,2), (3,3,2,1,1). Our proof of this result for each n ≥ 4 provides an explicit modulus M such that the corresponding equation has no solution already modulo M. This contributes to a classical problem posed by T. Skolem in 1930's on a local-global principle on purely exponential Diophantine equations.

Citations

• Solving an infinite number of purely exponential Diophantine equations with four terms
• Number of solutions to a special type of unit equations in two unknowns, III
• Number of solutions to a^x+b^y=c^z with gcd(a,b)>1
• Number of solutions to a special type of unit equations in two unknowns, II
• Powers of 3 with few nonzero bits and a conjecture of Erdős
• A Hasse-type principle for exponential diophantine equations and its applications

Related

• Solving an infinite number of purely exponential Diophantine equations with four terms
• On some conjectures of exponential Diophantine equations
• Skolem's conjecture confirmed for a family of exponential equations, II
• A diophantine equation for sums of consecutive like powers
• An exponential Diophantine equation related to the difference between\n powers of two consecutive Balancing numbers