1
vote
0answers
57 views

When is $n!+1$ a square? [duplicate]

I'm looking for the solutions $(n,m)$ of the equation $n!+1=m^2$. I have calculated the values of $\sqrt{n!+1}$ for $n \le $ and found only the solutions $(4,5)$, $(5,11)$ and $(7,71)$. Are these ...
3
votes
2answers
149 views

Factorials and Arithmetic Progression.

Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression. $n$ is a natural number I don't see any such examples(Except for $n=2$). And I don't see ...
18
votes
1answer
451 views

Integer solutions of $x! = y! + z!$

There was an interesting problem asked about triples $(x,y,z)$ which are solutions of $$x! = y! + z!.$$ Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$. Now I ...
3
votes
3answers
447 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?