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 $x∕y+y∕z+z∕x=4k^2$ does not have solutions in positive integers if $3\nmid k$. However, Garaev (Diophantine equations of the third degree, Tr. Mat. Inst. Steklova 218 (1997), 99–108) had already proved that the equation $x^3+y^3+z^3=nxyz$ has no positive integer solutions if $n=4k$, $n=8k-1$, or $n=2^{2m+1}(2k-1)+3$, where $m,k\in {\mathbb{Z}}^{+}$, which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation $x∕y+y∕z+m\cdot (z∕x)=nm$ does not have solutions in positive integers if $m$ is odd and $4\mid n$ or $8\mid n+1$. Our method is different from Garaev’s method and has been successfully applied to several situations.