Finite sets containing near-primitive roots

2020/06/26 by Agrawal, Komal, Pollack, Paul
#FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 11N25

paper · doi:10.48550/arxiv.2006.15200

Abstract

Fix a ∈ ℤ, a∉ \0,± 1\. A simple argument shows that for each ε> 0, and almost all (asymptotically 100% of) primes p, the multiplicative order of a modulo p exceeds p^\frac12-ε. It is an open problem to show the same result with \frac12 replaced by any larger constant. We show that if a,b are multiplicatively independent, then for almost all primes p, one of a,b,ab, a²b, ab² has order exceeding p^{(1)/(2)+(1)/(30)}. The same method allows one to produce, for each ε> 0, explicit finite sets A with the property that for almost all primes p, some element of A has order exceeding p^1-ε. Similar results hold for orders modulo general integers n rather than primes p.

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