# Journal of the Ramanujan Mathematical Society

Volume 35, Issue 3, September 2020 pp. 227–240.

Exponential Diophantine
equations pm - pn = qs - qt*

**Authors**:
Qingzhong Ji and Hourong Qin

**Author institution:**Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China

**Summary: **
Let q < p be two primes. We study the exponential Diophantine
equations pm - pn = qs - qt where m, n, s, t are positive
integers. In this paper, we prove that the equation
3m - 3n = 2s - 2t with m>n has only three solutions (m, n, s,
t)=(2, 1, 3, 1), (3, 1, 5, 3) and (5, 1, 8, 4); the equation
5m - 5n = 2s - 2t has only one solution (3, 1, 7, 3) and the
equation 13m - 13n = 3s - 3t has only one solution (3, 1, 7, 1).

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