A proof of the conjecture of Cohen and Mullen on sums of primitive roots

Abstract

We prove that for all q > 61, every non-zero element in the finite field F q can be written as a linear combination of two primitive roots of F q . This resolves a conjecture posed by Cohen and Mullen.

On consecutive primitive elements in a finite field

“…When n = 3 this gives the enormous bound 10 43 743 . The main point of this article is to apply techniques from [5] to prove the following result.…”

Primitive values of quadratic polynomials in a finite field

“…We now consider 1 ≤ ω(q − 1) ≤ 8 following the procedure in §2 of [6]. Consider ω(q − 1) = 8: there is no value of s ∈ [1,7] for which (10) is true.…”

Linear combinations of primitive elements of a finite field

“…Let F q denote the finite field of order q, a power of the prime p. The proliferation of primitive elements of F q gives rise to many interesting properties. For example, it was proved in [4] that for any non-zero α, β, ∈ F q the equation = aα + bβ is soluble in primitive elements a, b provided that q > 61. Since a is primitive if and only if a −1 , its multiplicative inverse in F q , is primitive, one may look for linear relations amongst primitive elements and their inverses and, as in the above example, seek a lower bound on q beyond which such relations hold -this is the purpose of the current paper.…”

“…In this paper we use a sieving method and some computation to establish the following theorem. 3,4,5,7,9,11,13,19,25,29,31,37,41,43,49,61,81,97, 121, 169}, E S = {2, 3, 4, 5, 7, 13}.…”