Suppose, $n\ge 8$ is an integer.
- Let $s$ be the smallest non-negative integer such that $n!-s$ is a perfect power.
- Let $t$ be the smallest non-negative integer such that $n!+t$ is a perfect power.
Conjecture : $\ n!-s\ $ and $\ n!+t\ $ are both perfect squares, but not perfect powers with an exponent larger than $2$. In other words, the nearest perfect powers are just $\ \lfloor \sqrt{n!} \rfloor ^2\ $ and $\ \lceil \sqrt{n!}\ \rceil ^2\ $.
The following PARI/GP program shows that the conjecture holds upto $\ n=1\ 000\ $.
gp > p=1;for(n=2,1000,p=p*n;d=p-sqrtint(p)^2;for(a=3,ceil(log(p)/log(2)),if(p-sqrtnint(p,a)^a<=d,print([n,a]))))
[4, 4]
[7, 3]
gp > p=1;for(n=2,1000,p=p*n;d=(sqrtint(p)+1)^2-p;for(a=3,ceil(log(p)/log(2)),if((sqrtnint(p,a)+1)^a-p<=d,print([n,a]))))
[3, 3]
[6, 3]
[6, 6]
gp >
Can we prove this conjecture ?