On Artin's primitive root conjecture and some related problems

Pieter Moree (Amsterdam)

The multiplicative order modulo m of a number n that is coprime to m is the smallest number k>0 such that n^k=1(mod m). We have k|phi(m) (Euler's totient function). If k=phi(m), then n is said to be a primitive root mod m. Artin's primitive root conjecture (1927) asserts that if g is not equal to -1 or a square, then there are infinitely many primes p for which g is a primitive root. We discuss the present status of this conjecture and some of its applications. Another problem concerning the multiplicative order we address is: how often is the order even ? This problem is closely related with that of prime divisors of integers determined by second order recurrences (such as Fibonacci and Lucas numbers). Some of my own results to be presented are joint work with Prof. P. Stevenhagen (Leiden University).

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