Melvyn Knight's Problem

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Melvyn Knight once asked [1] for which integers is there an integer triple so that .


For example, 11 has two such representations: and .

Solutions for this problem can be found using elliptic curves [1]. This Demonstration shows a sample solution for all solvable values from to . When one solution exists, there are an infinite number of solutions.


Contributed by: Ed Pegg Jr (June 13)
Open content licensed under CC BY-NC-SA


The original problem [1]

may be rewritten as


and further rewritten as


Solving for leads to a complicated expression equivalent to the elliptic curve:


A tabulation of the integers that can be represented is given in [2].


[1] A. Bremner, R. K. Guy and R. J. Nowakowski, "Which Integers Are Representable as the Product of the Sum of Three Integers with the Sum of Their Reciprocals?," Mathematics of Computation, 61(203), 1993 pp. 117–130.

[2] H. Pfoertner. "Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals." The On-Line Encyclopedia of Integer Sequences. (Jan 24, 2023)


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