Integers Representable as Product of both 3 and 4 Consecutive Integers
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Theorem
There are $3$ integers which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$:
- $24, 120, 175 \, 560$
Proof
We have:
\(\ds 24\) | \(=\) | \(\ds 1 \times 2 \times 3 \times 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 4\) |
\(\ds 120\) | \(=\) | \(\ds 2 \times 3 \times 4 \times 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 5 \times 6\) |
\(\ds 175 \, 560\) | \(=\) | \(\ds 55 \times 56 \times 57\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 19 \times 20 \times 21 \times 22\) |
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Sources
- 1972: David W. Boyd and Hershy Kisilevsky: The Diophantine equation u(u + 1)(u + 2)(u + 3) = v(v + 1)(v + 2) (Pacific J. Math. Vol. 40, no. 1: pp. 23 – 32)
- 1994: N. Saradha, T.N. Shorey and R. Tijdeman: On arithmetic progressions with equal products (Acta Arith. Vol. 68, no. 1: pp. 89 – 100)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $175,560$