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 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 does not have solutions in positive integers if . However, Garaev (Diophantine equations of the third degree, Tr. Mat. Inst. Steklova 218 (1997), 99–108) had already proved that the equation has no positive integer solutions if , , or , where , which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation does not have solutions in positive integers if is odd and or . Our method is different from Garaev’s method and has been successfully applied to several situations.
"On a remark of Sierpínski." Rocky Mountain J. Math. 52 (2) 717 - 726, April 2022. https://doi.org/10.1216/rmj.2022.52.717