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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.

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