Does a factorial always differ by a square from a square?

Question

Something I have noticed:

$$ 4!+1=5^2\\ 5!+1=11^2\\ 6!+3^2=27^2\\ 7!+1=71^2\\ 8!+9^2=201^2 $$

And you can go on. What is going on?

Answer 1

Hint: Note that $a^2-b^2=(a+b)(a-b)$.

Additional hint: Have a look at $2!$ or $3!$

Answer 2

HINT

We are claiming that

$$n!=a^2-b^2=(a+b)(a-b)$$

then consider $n!=rs$ and let

• $a+b=r$
• $a-b=s$