Journal of the Ramanujan Mathematical Society

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Quadratic Non-Residues Versus Primitive Roots Modulo p


Affiliations
1 Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico
2 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
3 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
     

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Given any ε ∈ (0, 1/2) and any positive integer s ≥ 2, we prove that for every prime

p ≥ max{s2(4/ε)2s, s651s log log(10s)} 

satisfying ϕ(p − 1)/(p − 1) ≤ 1/2 − ε, where ϕ(k) is the Euler function, there are s consecutive quadratic non-residues which are not primitive ischolar_mains modulo p.


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  • Quadratic Non-Residues Versus Primitive Roots Modulo p

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Authors

Florian Luca
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, Mexico
Igor E. Shparlinski
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
R. Thangadurai
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract


Given any ε ∈ (0, 1/2) and any positive integer s ≥ 2, we prove that for every prime

p ≥ max{s2(4/ε)2s, s651s log log(10s)} 

satisfying ϕ(p − 1)/(p − 1) ≤ 1/2 − ε, where ϕ(k) is the Euler function, there are s consecutive quadratic non-residues which are not primitive ischolar_mains modulo p.