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A proof of the conjecture of Cohen and Mullen on sums of primitive roots. (English) Zbl 1372.11113

Summary: We prove that for all \(q>61\), every non-zero element in the finite field \(\mathbb{F}_q\) can be written as a linear combination of two primitive roots of \(\mathbb{F}_q\). This resolves a conjecture posed by S. D. Cohen and G. L. Mullen [Appl. Algebra Eng. Commun. Comput. 2, No. 1, 45–53 (1991); erratum ibid. 2, No. 4, 297–299 (1991; Zbl 0744.11061)].

MSC:

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
11Y16 Number-theoretic algorithms; complexity

Citations:

Zbl 0744.11061
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References:

[1] Chou, Wun Seng; Mullen, Gary L.; Shiue, Jau-Shyong; Sun, Qi, Pairs of primitive elements modulo \(p^l\), Sichuan Daxue Xuebao, 26, Special Issue, 189-195 (1989) · Zbl 0701.11073
[2] Cohen, Stephen D., Pairs of primitive roots, Mathematika, 32, 2, 276-285 (1986) (1985) · Zbl 0581.10021 · doi:10.1112/S0025579300011050
[3] Cohen, Stephen D., Primitive elements and polynomials: existence results. Finite Fields, Coding Theory, and Advances in Communications and Computing, Las Vegas, NV, 1991, Lecture Notes in Pure and Appl. Math. 141, 43-55 (1993), Dekker, New York
[4] Cohen, Stephen D.; Huczynska, Sophie, The strong primitive normal basis theorem, Acta Arith., 143, 4, 299-332 (2010) · Zbl 1219.11182 · doi:10.4064/aa143-4-1
[5] Cohen, Stephen D.; Mullen, Gary L., Primitive elements in finite fields and Costas arrays, Appl. Algebra Engrg. Comm. Comput., 2, 1, 45-53 (1991) · Zbl 0744.11061 · doi:10.1007/BF01810854
[6] Golomb, Solomon W., Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37, 1, 13-21 (1984) · Zbl 0547.05020 · doi:10.1016/0097-3165(84)90015-3
[7] Guy, Richard K., Unsolved Problems in Number Theory, Problem Books in Mathematics, xviii+437 pp. (2004), Springer-Verlag, New York · Zbl 1058.11001
[8] Sun, Qi, On primitive roots in a finite field, Sichuan Daxue Xuebao, 25, 2, 133-139 (1988) · Zbl 0703.11074
[9] Szalay, Michael, On the distribution of the primitive roots of a prime, J. Number Theory, 7, 184-188 (1975) · Zbl 0302.10041
[10] Vegh, Emanuel, A note on the distribution of the primitive roots of a prime, J. Number Theory, 3, 13-18 (1971) · Zbl 0211.37202
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