$n! =$ the product of consecutive integers.

Question

Can $n!$ be the product of $k$ consecutive integers for $k > 1$? (Not including the degenerate cases such as when $k = 2$, then $1\cdot2 = 2!$ and $2\cdot 3 = 3!$, and so on.)

I am asking not for $n!$ to be divisible by $n$ consecutive integers, I am asking for $n!$ to equal the product of $k$ consecutive integers, implying that $n$ is not necessarily equal to $k$ (And when $n = k$, then clearly there exists an answer, namely $n! = (1)(2)(3)...(k)$)

$n!=1\cdot 2\cdot 3\cdot ...\cdot n$ is defined so as to be the product of $n$ consecutive integers. — walcher, Sep 02 '13 at 18:22
$n!$ can be the product of $n$ consecutive integers as well as $n-1$ consecutive integers. — Tomas, Sep 02 '13 at 18:22
Also $(k-1)!$ can be written as product of $k!-k$ integers in the following way: $(k+1)\cdot\dots\cdot k!$. — Tomas, Sep 02 '13 at 18:26
6! = 720 can be written as the product of three other consecutive integers: 8910 Is that what you mean? — imranfat, Sep 02 '13 at 18:38

score 7 · Answer 1 · answered Sep 02 '13 at 19:00

7

The formula $$(n!-1)!=\frac{(n!)!}{n!}=\prod_{k=n+1}^{n!} k$$ gives an infinite family of examples.

answered Sep 02 '13 at 19:00

Rebecca J. Stones

26,845

$n! =$ the product of consecutive integers.

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