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Question 5

Prove that the product of four consecutive positive integers cannot be equal to the product of two consecutive positive integers.

So it must equal $n(n+1)(n+2)(n+3)$ hence it must be divisible by 24 (In the sequence there must be a factor of 4,2,3. This must equal to $k(k+1)$. As $k$ and $k+1$ are co prime either $k$ or $k+1$ is divisible by 24, or one is divisible by 8 and the other by 3.

I run out of ideas since nothing pops out to me and the factorisations don't seem reveling. I also recognised the two formulae as 2*triangular numbers and 24*sum of the sum of triangular numbers, both appearing in pascals triangle. But thats more of a interesting observation then something useful in a proof.

I would appreciate a pointer how to proceed.

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  • $\begingroup$ Maybe try that $4k(k+1)+1$ is a perfect square? $\endgroup$ – Thomas Andrews Nov 27 '14 at 19:22
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$$n(n+1)(n+2)(n+3) = (n^2 + 3n)(n^2 + 3n +2)$$

If $n^2 + 3n \geq k$, then $(n^2 + 3n)(n^2 + 3n +2) > k(k+1)$

If $n^2 + 3n <k$, then $(n^2 + 3n)(n^2 + 3n +2) < k(k+1)$

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$$(n^2+3n)(n^2+3n+1)\lt n(n+1)(n+2)(n+3)\lt(n^2+3n+1)(n^2+3n+2)$$

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