Infinitely many correct solutions to equation?

Question

I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below:

$4!+1=5^2$

$5!+1=11^2$

$7!+1=71^2$

and so on...

Could I have a proof or disproof of my conjecture? (I don't even know where to start)

unfortunetly this was conjectured much before ,see Brocard conjectures — Booldy, Aug 18 '15 at 01:41

score 0 · Accepted Answer · answered Aug 18 '15 at 01:41

0

This is known as Brocard's problem.

It has been shown there are only finitely many solutions given the ABC conjecture is true (this I believe was proved recently), and it is conjectured the 3 solutions you have given are the only solutions. Calculations up to $n=10^9$ show no other solutions.

answered Aug 18 '15 at 01:41

Bennett Gardiner

4,163

Oh ok good to know – Aug 18 '15 at 01:42

Infinitely many correct solutions to equation?

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