Integer solutions to $x^x=122+231y$ How can I find the integer solutions to the following equation (without a script or trial and error)?
$$x^x=122+231y$$

By the Chinese Remainder Theorem, to know the value of $x^x$ modulo $231$ we only need to know it modulo $3,7,11$.
The value of $x^y$ modulo $n$ depends on the values of $x$ modulo $n$ and $y$ modulo $\phi(n)$.
So if $n$ is prime, the value of $x^x$ modulo $n$ depends on $x$ modulo $n(n-1)$.
A computation gives that
$x^x \equiv 122 \pmod 3 \iff x^x \equiv 2 \pmod 3 \iff x \equiv 5 \pmod 6$,
$x^x \equiv 122 \pmod 7 \iff x^x \equiv 3 \pmod 7 \iff x \equiv 5,31 \pmod {42}$
So far we know that we must have $x \equiv 5 \pmod {42}$.
Out of the $28$ solutions modulo $110$ to $x^x \equiv 122 \pmod {11}$, only $9$ are congruent to $1$ modulo $2$, and those are $1,5,15,23,25,45,67,75,89$ modulo $110$.
So modulo $2310$, we have $9$ solutions to $x^x = 122 \pmod {231}$, and they are $x \equiv 5,89,551,845,1013,1475,1895,1937,2105 \pmod {2310}$
