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April 2021 On the equations mn=1x±1y±1z
Sadegh Nazardonyavi
Rocky Mountain J. Math. 51(2): 669-688 (April 2021). DOI: 10.1216/rmj.2021.51.669

Abstract

We study the equations mn=1x±1y±1z, where m=4,5,6,7 and n>1 is a given positive integer. Except for the case mn=1x+1y+1z (i.e., all signs are positive), which are known conjectures of Erdős–Straus (m=4), Sierpiński (m=5) and Schinzel (m1), we show that the equations (with at least one minus sign) have solutions x,y,z, where z>y>x>0 for sufficiently large n.

Citation

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Sadegh Nazardonyavi. "On the equations mn=1x±1y±1z." Rocky Mountain J. Math. 51 (2) 669 - 688, April 2021. https://doi.org/10.1216/rmj.2021.51.669

Information

Received: 15 July 2020; Revised: 26 September 2020; Accepted: 29 September 2020; Published: April 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.1216/rmj.2021.51.669

Subjects:
Primary: 11DXX , 11-xx , 11Y50

Keywords: Diophantine equation , Egyptian fraction , Schinzel conjecture , unit fraction

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 2 • April 2021
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