Product of $5$ consecutive integers cannot be perfect square

Question

How can we prove that the product of $5$ consecutive positive integers cannot be a perfect square?

Answer 1

I see no need to retype the answer given here, which is the first result when putting the title of this question into Google.

Answer 2

This paper of Erdős proves that the product of two or more consecutive numbers is never a square. This question was also asked before on Math.SE. You can also relate to Math Forum or google "product of five consecutive numbers is never a square" and variants to get many interesting results.

Answer 3

Let $N=n(n+1)(n+2)(n+3)(n+4)$. Then it is easily seen that $N=2^a3^b5^{2k_1}7^{2k_2}\cdots$ where $a\geq 2,b\geq 1, k_1\geq 1$ and $k_2,k_3,\dots\geq 0$.

We consider four possible forms of $n+i,0\leq i\leq 4$:

If $2| a,b$ then let $a,b=2m_1,2m_2$ so that $n+i=(2^{m_1}3^{m_2}5^{k_1}\dots)^2$

If $2|a, 2\nmid b$ then let $a,b=2m_1,2m_2+1$ so that $n+i=3(2^{m_1}3^{m_2}5^{k_1}\dots)^2$

Continuing, we find $2\nmid a, 2|b\implies n+i=2(2^{m_1}3^{m_2}5^{k_1}\dots)^2$ and $2\nmid a,b\implies n+i=6(2^{m_1}3^{m_2}5^{k_1}\dots)^2$

By the pigeonhole principle, at least two forms will occur in $N$. Clearly, the two forms bust have $2|a,b$ which implies that the sequence contains two squares. But the only sequence which does this is ${1,2,3,4,5}$ which does not multiply to a product. Thus, no five consecutive integers can multiply to a perfect square.