Small prime kth power residues for k=2,3,4: A reciprocity laws\n approach

2017/11/05 by Kübra Benli̇, Paul Pollack, Benli, Kübra +1
Mathematics · #11N36 (secondary) #1A15 (primary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1711.01706

openalex publication_date 2017/11/05 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Nagell proved that for each prime p\≡ 1 pmod3, p > 7, there is a\nprime q<2p^1/2 that is a cubic residue modulo p. Here we show that for\neach fixed \ε > 0, and each prime p\≡ 1 pmod3 with p >\np₀(\ε), the number of prime cubic residues q < p^1/2+\ε\nexceeds p^\ε/30. Our argument, like Nagell's, is rooted in the law of\ncubic reciprocity; somewhat surprisingly, character sum estimates play no role.\nWe use the same method to establish related results about prime quadratic and\nbiquadratic residues. For example, for all large primes p, there are more\nthan p^1/9 prime quadratic residues q<p.\n

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