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

Abstract

We prove that for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than 61"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>61</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q&gt;61</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , every non-zero element in the finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be written as a linear combination of two primitive roots of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This resolves a conjecture posed by Cohen and Mullen.