Permutations of even residues modulo p
by
Jennifer Paulhus
Kansas State University
Coauthors: Todd Cochrane Christopher Pinner

Abstract: Let p be an odd prime with A, d positive integers satisfying (d, p-1)=1 and p not dividing A. Goresky and Klapper conjecture that for p > 13 with 2 primitive modulo p, if the map sending x → Ax^d permutes the even residues modulo p then A=d=1. This conjecture arises from an equivalent conjecture about a class of binary sequences called l-sequences. These l-sequences may be thought of as the 2-adic expansion of a rational number r/q with (r, q)=1, the output for feedback shift registers with carry, or a codeword in the Barrows-Mandelbaum arithmetic code. We use binomial sum bounds to prove the Goresky-Klapper conjecture for any prime p > 4 ·10⁹⁰.

Date received: June 15, 2008