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Let  a  be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that  # { p : ℓ   a  ( p )= r }≪ ε  r  ε  for any ε>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 .

ON A CONJECTURE OF ERDŐS