Smallest Number which is Sum of 4 Triples with Equal Products

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Theorem

The smallest positive integer which is the sum of $4$ distinct ordered triples, each of which has the same product, is $118$:

\(\displaystyle 118\) \(=\) \(\displaystyle 14 + 50 + 54\)
\(\displaystyle \) \(=\) \(\displaystyle 15 + 40 + 63\)
\(\displaystyle \) \(=\) \(\displaystyle 18 + 30 + 70\)
\(\displaystyle \) \(=\) \(\displaystyle 21 + 25 + 72\)


Proof

\(\displaystyle 14 \times 50 \times 54\) \(=\) \(\displaystyle \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle 37 \, 800\)


\(\displaystyle 15 \times 40 \times 63\) \(=\) \(\displaystyle \paren {3 \times 5} \times \paren {2^3 \times 5} \times \paren {3^2 \times 7}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle 37 \, 800\)


\(\displaystyle 18 \times 30 \times 70\) \(=\) \(\displaystyle \paren {2 \times 3^2} \times \paren {2 \times 3 \times 5} \times \paren {2 \times 5 \times 7}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle 37 \, 800\)


\(\displaystyle 21 \times 25 \times 72\) \(=\) \(\displaystyle \paren {3 \times 7} \times \paren {5^2} \times \paren {2^3 \times 3^2}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle 37 \, 800\)



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