Consecutive Quadratic Residues and Quadratic Nonresidue Modulo P
Consecutive Quadratic Residues And Quadratic Nonresidue Modulo p N. A. Carella Abstract: Let p be a large prime, and let k log p. A new proof of the existence of ≪ any pattern of k consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues np modulo p shows that n (log p)(log log p). p ≪ 1 Introduction Given a prime p 2, a nonzero element u F is called a quadratic residue, equiva- ≥ ∈ p lently, a square modulo p whenever the quadratic congruence x2 u 0mod p is solvable. − ≡ Otherwise, it called a quadratic nonresidue. A finite field Fp contains (p + 1)/2 squares = u2 mod p : 0 u < p/2 , including zero. The quadratic residues are uniformly R { ≤ } distributed over the interval [1,p 1]. Likewise, the quadratic nonresidues are uniformly − distributed over the same interval. Let k 1 be a small integer. This note is concerned with the longest runs of consecutive ≥ quadratic residues and consecutive quadratic nonresidues (or any pattern) u, u + 1, u + 2, . , u + k 1, (1) − in the finite field F , and large subsets F . Let N(k,p) be a tally of the number of p A ⊂ p sequences 1. Theorem 1.1. Let p 2 be a large prime, and let k = O (log p) be an integer. Then, the ≥ finite field Fp contains k consecutive quadratic residues (or quadratic nonresidues or any pattern). Furthermore, the number of k tuples has the asymptotic formulas p 1 k 1 (i) N(k,p)= 1 1+ O , if k 1.
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