What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)?

Tho, Nguyen Xuan (2021) What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)? Annales Mathematicae et Informaticae, 54. pp. 141-146. ISSN 1787-6117

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Abstract

This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.

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