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 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:

\(\displaystyle 510 \, 510\) \(=\) \(\displaystyle 13 \times 21 \times 34 \times 55\)
\(\displaystyle \) \(=\) \(\displaystyle 13 \times \paren {3 \times 7} \times \paren ({2 \times 17} \times \paren {5 \times 11}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17\)

$\blacksquare$


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