ON A CONJECTURE OF ERDOS

Let a be an integer different from 0, +/-1, or a perfect square. We consider a conjecture of Erdos which states that #{p : l(a)(p) = r} <<(epsilon) r(epsilon) for any epsilon > 0, where l(a)(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin's primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p + a and the order of a modulo p.