# Melvyn Knight's Problem

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

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Contributed by: Ed Pegg Jr (June 13)

Open content licensed under CC BY-NC-SA

## Details

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].

References

[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. www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1189516-5/S0025-5718-1993-1189516-5.pdf.

[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) oeis.org/A085514.

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