# solving a Diophantine equation through respective congruences?

Question:

1-Let $$a,b$$ be integers having the property that for any prime power $$t$$ there exists an integer $$v_t$$ such that $$b\equiv a^{v_t}\pmod t.$$ Then there exists an integer $$v$$ such that $$b=a^v.$$

2-Does it hold with $$t$$ being primes not prime powers?

thanks!

• Hensel's Lemma might help – Dr. Mathva Apr 7 at 15:34
• What are your thoughts on the question? What have you tried and where did you get stuck? Is there more context to this problem? Did you find it in a book? Do you know Hensel's lemma? – Servaes Apr 7 at 16:39
• @Servaes, it is a theorem in a book about sieve methods, but I would like to know if there are more elementary way to solve it. – asad Apr 7 at 20:00