# Expected number of digits of the smallest prime factor of $77^{77}-18$

Let $$X$$ be the number of digits of the smallest prime factor of $$77^{77}-18$$ which is a composite $$146$$-digit number. ECM indicates that the smallest factor has more than $$30$$ digits.

Assuming that no prime factor with $$30$$ digits or less exists, how can I calculate $$E(X)$$ ?

I am aware of the estimation of the number of $$y$$-rough numbers below $$x$$, but I think this approach is not very accurate for such a relatively small number.