April 2022 On a remark of Sierpínski
Nguyen Xuan Tho
Rocky Mountain J. Math. 52(2): 717-726 (April 2022). DOI: 10.1216/rmj.2022.52.717


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 xy+yz+zx=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 xy+yz+zx=4k2 does not have solutions in positive integers if 3k. 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=8k1, or n=22m+1(2k1)+3, where m,k+, which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation xy+yz+m(zx)=nm does not have solutions in positive integers if m is odd and 4n or 8n+1. Our method is different from Garaev’s method and has been successfully applied to several situations.


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Nguyen Xuan Tho. "On a remark of Sierpínski." Rocky Mountain J. Math. 52 (2) 717 - 726, April 2022. https://doi.org/10.1216/rmj.2022.52.717


Received: 22 April 2021; Revised: 17 August 2021; Accepted: 25 August 2021; Published: April 2022
First available in Project Euclid: 17 May 2022

Digital Object Identifier: 10.1216/rmj.2022.52.717

Primary: 11D68
Secondary: 11D25 , 11D72

Keywords: Elliptic curves , Hilbert symbols , sums of positive rationals

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.52 • No. 2 • April 2022
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