Smallest Integer which is Product of 4 Triples all with Same Sum
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:
\(\ds 6 \times 56 \times 75\) | \(=\) | \(\ds \paren {2 \times 3} \times \paren {2^3 \times 7} \times \paren {3 \times 5^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^2 \times 5^2 \times 7\) | ||||||||||||
\(\ds 6 + 56 + 75\) | \(=\) | \(\ds 137\) |
\(\ds 7 \times 40 \times 90\) | \(=\) | \(\ds 7 \times \paren {2^3 \times 5} \times \paren {2 \times 3^2 \times 5^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^2 \times 5^2 \times 7\) | ||||||||||||
\(\ds 7 + 40 + 90\) | \(=\) | \(\ds 137\) |
\(\ds 9 \times 28 \times 100\) | \(=\) | \(\ds 3^2 \times \paren {2^2 \times 7} \times \paren {2^2 \times 5^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^2 \times 5^2 \times 7\) | ||||||||||||
\(\ds 9 + 28 + 100\) | \(=\) | \(\ds 137\) |
\(\ds 12 \times 20 \times 105\) | \(=\) | \(\ds \paren {2^2 \times 3} \times \paren {2^2 \times 5} \times \paren {3 \times 5 \times 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^2 \times 5^2 \times 7\) | ||||||||||||
\(\ds 12 + 20 + 105\) | \(=\) | \(\ds 137\) |
This theorem requires a proof. In particular: It remains to be shown that this is the smallest such number with this property. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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
- Feb. 1981: J.G. Mauldon: Elementary Problems: E2872 (Amer. Math. Monthly Vol. 88, no. 2: p. 148) www.jstor.org/stable/2321140
- Sep. 1982: Lorraine L. Foster and Gabriel Robins: E2872 (Amer. Math. Monthly Vol. 89, no. 7: pp. 499 – 500) www.jstor.org/stable/2321396
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $25,200$
- 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.): $\text D 16$: Triples with the same sum and same product