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 $118$:

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


Proof

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


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


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


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


Computational proof

The following python script finds the maximum number of ways to represent each positive integer as sums of triples with the same product: 

def SearchTriplets(n):
    rmap = {}
    # iterate i up to i + 1 + 1 = n
    for i in range(1,n-2):
        # iterate j up to i + j + 1 = n
        for j in range(i,n-i-1):
            k = n - i - j
            vec = [i,j,k]
            # sort list to make unique representation
            vec.sort()
            p1 = i*j*k
            # Add or insert vec to product dictionary if it does not exist
            if p1 in rmap:
                if not vec in rmap[p1]:
                     rmap[p1].append(vec)
            else:
                rmap[p1] = [ vec ]
    maxv = []
    products = list(rmap.keys())
    # report longest list - with lowest product if multiple list with equal lengths exist
    products.sort()
    for p in products:
        v = rmap[p]
        if len(v) > len(maxv):
            maxv = v
    return maxv

small = 1
for i in range(1000):
    res = SearchTriplets(i)
    if len(res) > small:
        print(i, len(res), res)
        small = len(res)

This outputs: 

13 2 [[1, 6, 6], [2, 2, 9]]
39 3 [[4, 15, 20], [5, 10, 24], [6, 8, 25]]
118 4 [[14, 50, 54], [15, 40, 63], [18, 30, 70], [21, 25, 72]]
185 5 [[11, 84, 90], [12, 63, 110], [15, 44, 126], [18, 35, 132], [22, 28, 135]]
400 6 [[24, 180, 196], [27, 128, 245], [28, 120, 252], [32, 98, 270], [36, 84, 280], [42, 70, 288]]
511 7 [[35, 216, 260], [36, 195, 280], [40, 156, 315], [42, 144, 325], [45, 130, 336], [60, 91, 360], [72, 75, 364]]

This sequence is A103277 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


This confirms the statement that $118$ is the smallest positive integer that satisfies the criterion.

$\blacksquare$


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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