Suppose $F_n$ to be the nth term of the Fibonacci sequence.($F_1=F_2=1,F_{n+2}=F_{n+1}+F_n$)
(1)Find all duals $(m,n)$, so that $$F_m^m=(F_n^n+1)^2$$. (2)Find all triples $(m,n,t)$, so that $$F_m^m=F_n^n(F_t^t+2)+1$$.
for (1):since $$F_{3}=2,F_{4}=3,F_{5}=5,F_{6}=8,\cdots,$$ if $m=2$,then $(F_{m})^m=1$,then not solution,because $RHS>1$
if $m=4$,then $9=F_{4}^2=(F_{n})^n+1$,then $n=3$,so $(m,n)=(4,3)$ is solution
if $m=6$,then $ 8^3=(F_{n})^n+1$, it is clear no solution
Gerry Myerson point that can use Catalan:since $F^n_{n}+1$ can't be a power for $n>3$.so $n>3$ the equation can't have solution
so for (2) problem maybe can't use this result?