On a remark of Sierpínski

Abstract

In a remark on page 80 of his classical book 250 Problems in Elementary Number Theory, Sierpiński stated that it was not known if the equation x∕y+y∕z+z∕x=4 has solutions in positive integers. Bondarenko (Investigation of a class of Diophantine equations, Ukraïn. Mat. Zh. 52:6 (2000), 831–836) gave a negative answer to Sierpiński’s remark by showing that the equation x∕y+y∕z+z∕x=4k2 does not have solutions in positive integers if 3∤k. However, Garaev (Diophantine equations of the third degree, Tr. Mat. Inst. Steklova 218 (1997), 99–108) had already proved that the equation x3+y3+z3=nxyz has no positive integer solutions if n=4k, n=8k−1, or n=22m+1(2k−1)+3, where m,k∈ℤ+, which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation x∕y+y∕z+m⋅(z∕x)=nm does not have solutions in positive integers if m is odd and 4∣n or 8∣n+1. Our method is different from Garaev’s method and has been successfully applied to several situations.