New York Journal of Mathematics
Volume 16 (2010) 387-398


Joseph Lewittes and Victor Kolyvagin

Primes, permutations and primitive roots

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Published: November 9, 2010
Keywords: permutation, prime, primitive root, class number
Subject: 11A07, 11R29, 20B35

Let p be a prime greater than 3, X = { 1, 2, ..., p-1 } and R the set of primitive roots mod p contained in X. To each g ∈ R associate the permutation σg of X defined by σg(x) = y where y is the unique member of X satisfying y ≡ gx mod p. Let ΣR = { σg | g ∈ R }. We analyze the parity of the permutations in ΣR. If p ≡ 1 mod 4 half the permutations are even and half are odd. If p ≡ 3 mod 4 they are either all even or all odd; set ε(p) = 1 in the even case, ε(p) = -1 in the odd case. Numerical evidence suggests the conjecture that ε(p) ≡ h(-p) mod 4, where h(-p) is the class number of the quadratic field Q(\sqrt{-p}). The conjecture is shown to be true, and furthermore ε(p) ≡ -((p-1/2))! mod p. We also study a larger class of permutations of degree p-1 which generalize the ΣR.

Author information

Joseph Lewittes:
Joseph Lewittes, Department of Mathematics and Computer Science, Lehman College - CUNY, 250 Bedford Park Boulevard West, Bronx, NY 10468

Victor Kolyvagin:
Victor Kolyvagin, The Graduate Center - CUNY, 365 Fifth Avenue, New York, NY 10016