Smallest Integer which is Product of 4 Triples all with Same Sum

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Theorem

The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is:

\(\ds 25 \, 200\) \(=\) \(\ds 6 \times 56 \times 75\)
\(\ds \) \(=\) \(\ds 7 \times 40 \times 90\)
\(\ds \) \(=\) \(\ds 9 \times 28 \times 100\)
\(\ds \) \(=\) \(\ds 12 \times 20 \times 105\)


Proof

We have:

\(\ds 6 \times 56 \times 75\) \(=\) \(\ds \paren {2 \times 3} \times \paren {2^3 \times 7} \times \paren {3 \times 5^2}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2 \times 5^2 \times 7\)
\(\ds 6 + 56 + 75\) \(=\) \(\ds 137\)


\(\ds 7 \times 40 \times 90\) \(=\) \(\ds 7 \times \paren {2^3 \times 5} \times \paren {2 \times 3^2 \times 5^2}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2 \times 5^2 \times 7\)
\(\ds 7 + 40 + 90\) \(=\) \(\ds 137\)


\(\ds 9 \times 28 \times 100\) \(=\) \(\ds 3^2 \times \paren {2^2 \times 7} \times \paren {2^2 \times 5^2}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2 \times 5^2 \times 7\)
\(\ds 9 + 28 + 100\) \(=\) \(\ds 137\)


\(\ds 12 \times 20 \times 105\) \(=\) \(\ds \paren {2^2 \times 3} \times \paren {2^2 \times 5} \times \paren {3 \times 5 \times 7}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2 \times 5^2 \times 7\)
\(\ds 12 + 20 + 105\) \(=\) \(\ds 137\)


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Also see


Historical Note

Richard K. Guy discusses this result in his Unsolved Problems in Number Theory of $1981$, and carries it forward into later editions.

In his Unsolved Problems in Number Theory, 3rd ed. of $2004$, the result is presented as:

It may be of interest to ask for the smallest sums or products with each multiplicity. For example, for $4$ triples, J. G. Mauldon finds the smallest common sum to be $118$ ... and the smallest common product to be $25200$ ...


However, in the article cited by Richard K. Guy, which appears in American Mathematical Monthly for Feb. $1981$, in fact J. G. Mauldon does no such thing.

Instead, he raises the question for $5$ such triples.


David Wells, in his Curious and Interesting Numbers, 2nd ed. of $1997$, propagates this, accrediting the result to Mauldron, citing that same problem in American Mathematical Monthly.

It is also apparent that Mauldron is a misprint for J.G. Mauldon.


Sources

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