510,510 is Product of 4 Consecutive Fibonacci Numbers
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Theorem
$510 \, 510$ can be expressed as the product of $4$ distinct consecutive Fibonacci numbers:
- $510 \, 510 = 13 \times 21 \times 34 \times 55$
and is also the $7$th primorial:
- $510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
Proof
By observation:
\(\ds 510 \, 510\) | \(=\) | \(\ds 13 \times 21 \times 34 \times 55\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 \times \paren {3 \times 7} \times \paren {2 \times 17} \times \paren {5 \times 11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17\) |
$\blacksquare$
Historical Note
David Wells report in his $1997$ work Curious and Interesting Numbers, 2nd ed. that this result can be found in an article by Monte James Zerger in Journal of Recreational Mathematics volume $12$, but this has not been corroborated.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $510,510$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $510,510$