Consecutive non-square non-primitive tuples in finite fields

2026/07/19 by Juncheng Zhou, Hongfeng Wu
#math.NT

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Abstract

Let q be an odd prime power and put θ_q=(φ(q-1))/(q-1). Let \mathbbF_q denote a finite field with q elements, an element of \mathbbF_q is called non-square non-primitive, or NSNP, if it is both a non-square and a non-primitive element. We first obtain a general existence theorem for consecutive tuples of non-square ℓth powers, where ℓ is an odd prime divisor of q-1. More precisely, if k≥ 2, char\mathbbF_q≥ k, and q>(k-1)²(2ℓ)^2k, then \mathbbF_q contains k consecutive non-square ℓth powers. Combining this result with a finite computation, we prove that θ_q<4/15 guarantees the existence of three consecutive NSNP elements. On the boundary θ_q=4/15, the only exceptions are \mathbbF₃₁, \mathbbF₆₁, \mathbbF₁₂₁. In particular, the constant 4/15 is best possible.

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