Solutions to Diophantine Equation x (x + 1) = y (y + 5) (y + 10) (y + 15)

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Theorem

The Diophantine equation

$n = x \left({x + 1}\right) = y \left({y + 5}\right) \left({y + 10}\right) \left({y + 15}\right)$

has exactly $2$ solutions:

\(\ds 1056\) \(=\) \(\ds 32 \times 33 = 1 \times 6 \times 11 \times 16\)
\(\ds 43 \, 056\) \(=\) \(\ds 207 \times 208 = 8 \times 13 \times 18 \times 23\)


Proof


Historical Note

According to David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$, this result appeared in Acta Arithmetica on page $194$ of volume $7$.

However, this contributor has tried to identify the article in question, and it appears to have nothing to do with the theorem discussed here.


Sources