COLLOQUIUM MATHEMATICUM VOL. 128 2012 NO. 2 ON THE DIOPHANTINE EQUATION x y - y x = c z BY ZHONGFENG ZHANG (Zhaoqing), JIAGUI LUO (Zhaoqing) and PINGZHI YUAN (Guangzhou) Abstract. Applying results on linear forms in p-adic logarithms, we prove that if (x, y, z) is a positive integer solution to the equation x y - y x = c z with gcd(x, y) = 1 then (x, y, z) = (2, 1,k), (3, 2,k), k ≥ 1 if c = 1, and either (x, y, z)=(c k +1, 1,k), k ≥ 1 or 2 ≤ x<y ≤ max{1.5 × 10 10 ,c} if c ≥ 2. 1. Introduction. Kenichiro Kashihara [Ka] asked to solve the equation x y + y z + z x = 0, i.e., to find its integer solutions. Recently, Yanni Liu and Xiaoyan Guo [LG] answered this question by showing that (x, y, z )= (-2, 1, 1), (1, -2, 1), (1, 1, -2), (1, -1, -2), (-1, -2, 1), (-2, 1, -1) are the only integer solutions. Let a, b, c be odd positive integers and H = max{a, b, c}. The first and the third authors [ZY] proved that all integer solutions to the equation ax y + by z + cz x = 0 with xyz 6= 0 satisfy max{|x|, |y|, |z |} ≤ 2H . In that paper, they also considered the equation x y + y z = z x and using a result of Stewart and Kunrui Yu [SY] on the ABC conjecture, showed that all positive integer solutions satisfy max{x, y, z } < exp(exp(exp(5))). The aim of this paper is to consider the equation x y - y x = c z , where the positive integer c is given, and to prove the following result. Theorem 1.1. Let c be a positive integer, and (x, y, z ) a positive integer solution to the equation (1.1) x y - y x = c z , gcd(x, y)=1. Then either (i) c =1, (x, y, z ) = (2, 1,k), (3, 2,k), k ≥ 1; or (ii) c ≥ 2, (x, y, z )=(c k +1, 1,k), k ≥ 1, or 2 ≤ x<y ≤ max{1.5 × 10 10 ,c}. In the case where c ≥ 2 and y>x ≥ 2, from c z <x y we get z< y log x/log c<y log y/log c, which is a bound for z depending on c. We deduce the following corollary. 2010 Mathematics Subject Classification : Primary 11D61; Secondary 11D41. Key words and phrases : exponential diophantine equation, linear forms in logarithms. DOI: 10.4064/cm128-2-13 [277] c  Instytut Matematyczny PAN, 2012