 

Joseph Lewittes and Victor Kolyvagin
Primes, permutations and primitive roots view print


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


Abstract
Let p be a prime greater than 3, X = { 1, 2, ..., p1 } and R the
set of primitive roots mod p contained in X.
To each g ∈ R associate the permutation σ_{g} of X defined by
σ_{g}(x) = y where y is the unique member of X satisfying
y ≡ g^{x} mod p.
Let Σ_{R} = { σ_{g}  g ∈ R }.
We analyze the parity of the permutations in Σ_{R}.
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) ≡ ((p1/2))! mod p.
We also study a larger class of permutations of degree p1 which generalize
the Σ_{R}.


Author information
Joseph Lewittes:
Joseph Lewittes, Department of Mathematics and Computer Science, Lehman College  CUNY, 250 Bedford Park Boulevard West, Bronx, NY 10468
joseph.lewittes@lehman.cuny.edu
Victor Kolyvagin:
Victor Kolyvagin, The Graduate Center  CUNY, 365 Fifth Avenue, New York, NY 10016
vkolyvagin@gc.cuny.edu

