With Bertrand's postulate at hand, it is easy to see that $n!$ is never a square for $n\ge 2$ (because there is a prime between $n/2$ and $n$).
But are there more elementary proofs of that fact?
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See whether this helps you: https://www.quora.com/What-is-a-proof-that-n-is-not-a-perfect-square-without-using-Bertrands-postulate – Dec 14 '17 at 14:25
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Well, I'd hardly say that that proof is more elementary. – ajotatxe Dec 14 '17 at 14:30
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Yes, you are right. – Dec 14 '17 at 14:30
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2See $n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem? – mathlove Dec 14 '17 at 14:53
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@mathlove This is near from what I'm looking for. But if you compaqre Erdős' proof of the Bertrands' postulate with this solution, you can note that they are almost identical. That is, this proof has an "embedded" proof the Bertrand's postulate. – ajotatxe Dec 15 '17 at 15:02
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@ajotatxe That is not true at all. The proof at that solution is much easier than the full proof of Bertrand's postulate. Indeed, the proof of Bertrand's postulate uses several other ideas besides the one used there. – John Gowers Jun 18 '18 at 09:19
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Not an answer just some speculative thoughts.
Let's define $T_n$ as the n'th tetrahedral number, where
$$T_n=\frac{1}{6}n(n+1)(n+2)$$
The even tetrahedral numbers $T_{2n}$ are also given by
$$T_{2n}=2^2+4^2+6^2+...+(2n)^2$$
The odd tetrahedral numbers $T_{2n-1}$ are also given by
$$T_{2n-1}=1^2+3^2+5^2+...+(2n-1)^2$$
We can define the factorial in terms of a tetrahedral number thus
$$n!=6(n-3)! \;T_{n-2}$$
We can then iterate this process as long as the residual factorial left over is 2! or greater. The factorial is then calculated from the product of two or more tetrahedral numbers, alternating between odd and even tetrahedral numbers.
$$n!=6^2(n-6)! \;T_{n-5}\;T_{n-2}$$ $$n!=6^3(n-9)! \;T_{n-8}\;T_{n-5}\;T_{n-2}$$
According to Dickson (History of the Theory of Numbers (Vol II. p.25)) A.J.J. Meyl [REF1] has proved that only three tetrahedral numbers are perfect squares; that is $T_1=1^2$, $T_2=2^2$ and $T_{48}=140^2$. Assuming it is correct it is an elementary proof.
The question is whether it is possible to build on this proof and the properties of tetrahedral numbers, to determine when the product of odd and even tetrahedral numbers are squares or not squares.
[REF1] Meyl, A.J.J. "Solution de Question 1194." Nouv. Ann. Math. 17, 464-467, 1878. http://www.numdam.org/article/NAM_1878_2_17__464_1.pdf
James Arathoon
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