Smallest Number which is Sum of 4 Triples with Equal Products
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
- 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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $118$
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $118$
- 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.): $\text D 16$: Triples with the same sum and same product