Let $a,b$ be integers , $a\ge 3$ , $b\ge 1$
If $f_a$ is the $a$ th cyclotomic polynomial, are there any perfect powers of the form $f_a(b)$ apart from $121$ and $343$ ?
In the range $$3\le a\le 1\ 000\ ,\ 1\le b\le 1\ 000$$ there are no more perfect powers as this pari/gp program shows :
gp > for(a=3,1000,f=polcyclo(a);for(b=1,1000,s=subst(f,x,b);if(ispower(s)>0,print([a,b,factor(s)]))))
[3, 18, Mat([7, 3])]
[5, 3, Mat([11, 2])]
[6, 19, Mat([7, 3])]
gp >
Any idea or reference ? Is it known whether only the $3$ pairs give a perfect power ?
Update : I extended the range to $a,b\le 2\ 000$ without finding another pair.