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)?
$$x^x=122+231y$$

We are attempting to solve:
$$ x^x \equiv 122 \  \mod \ 231  $$
Thus we naturally would want to factor 231 which is
$$ 3*7*11 $$
Thus we can break this into a system of 3 modular equations
$$ x^x \equiv 2 \mod \ 3$$
$$ x^x \equiv 3 \mod \ 7$$
$$ x^x \equiv 1 \mod \ 11$$
upon exploring the equation it appears there are solutions of the form respectively for
$$ x \equiv 5 \mod \ 6$$ (appears to solve the mod 3)
$$ x \equiv 5 \mod \ 84 $$ (appears to solve the mod 7)
And:
$$ x \equiv 1 \mod \ 11 $$ (solves the mod 11 but isn't the only case)
Why these in particular? I don't know...
Its worth noting
$$ x \equiv 0 \mod 5 $$ (the other apparent case)
Appears to imply
$$ x \equiv 1 \ or \ -1 \mod 11 $$  
Again: it appears now that:
$$ j*84*10^k + 5 $$ is constant for all k > 0 for fixed j
specifically if j = 0 or j = 1 then for all k > 0 this appears to be solution:
sample solutions include:
5,89, 845, 8405, 84005 etc...
