I'm not too sure, but I believe I first need to find a primitive root of $139$, but don't know if there is any better way asides from brute-forcing it and trying bases in the residue system (up to 138) such that $b^{138}\equiv 1 \pmod{139}$ as it is taking me forever since the mod is huge. But even if I have my primtitive root, I am stuck... All of the examples in the textbook I've seen so far dealt with much smaller mods and powers. Furthermore, both the power and mod are primes.
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1Hint: find $83^{-1}\pmod{138}$ and use Fermat's little theorem. – Greg Martin Oct 24 '23 at 04:31
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where does finding the primitive roots come in? (Or does it not need to be used? All of the examples I've seen use it) – Jason Xu Oct 24 '23 at 04:37
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1You don't need a primitive root. Were the examples you saw exactly like this one (prime base $p = 139$, exponent $83$ is coprime to $p-1$, the unknown appears in the base and not in the exponent)? – Hans Engler Oct 24 '23 at 04:45
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Apply the Theorem in the linked dupe. – Bill Dubuque Oct 24 '23 at 06:13