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Consecutive Quadratic Residues and Quadratic Nonresidue Modulo P

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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. 2k − p p ≥ arXiv:2011.11054v2 [math.GM] 28 Dec 2020 p (ii) N(k,p)= + O k2 , if k 1. 2k ≥ The first expression is suitable for applications requiring any integer k, and the second expression is suitable for applications requiring small integers k. The proofs are assembled in Section 8. The main results are proved using a new counting technique, based on Lemma 5.2. It provides sharper error terms than the standard technique based on Lemma 5.1. December 29, 2020 MSC2020 : Primary 11A15, Secondary 11L40. Keywords: Least quadratic nonresidue, Consecutive quadratic residues. 1 Consecutive Quadratic Residues And Quadratic Nonresidues Modulo p 2 Theorem 1.2. Let p 2 be a large prime, and let k = O (log p) be an integer. Then, ≥ for any subset of consecutive elements F of cardinality p1 ε/2 # contains k A ⊂ p − ≪ A consecutive quadratic residues (or quadratic nonresidues or any pattern), ε > 0 is an arbitrary small number. Furthermore, the number of k tuples has the asymptotic formulas # 1 k 1 (i) N(k,p, )= A 1 1+ O , if k 1. A 2k − p p ≥ # (ii) N(k,p, )= A + O (k) , if k 1. A 2k ≥ Quadratic residues r F (and quadratic non residues) in finite fields have orders ord (r)= ∈ p p 2. The analysis and results for k consecutive d power residues or any pattern of d power residues have similar details, but are more complex as the orders of the elements increases. The other result consider an interesting application to the least quadratic nonresidue. The other result considers an interesting application to the least quadratic nonresidue np modulo p. The current unconditional result for the least quadratic nonresidues in the literature states that n p1/4√e+ε, (2) p ≪ where ε> 0, and the strongest conditional result for primitive character χ states that n (p) (log p)1.37+o(1), (3) χ ≪ see [4, Corollary 2], and Conjecture 10.1. The following result is proved here. Theorem 1.3. For any large prime p 2, the least quadratic nonresidue is bounded by ≥ n (log p)(log log p). (4) p ≪ The implied constant should be small, perhaps 20, see Table 1 in Section 12. In Section ≤ 10 several Lemmas are spliced together to prove this result. Section 3 to Section ?? cover the supporting materials and other optional topics. 2 Quadratic Symbol Definition 2.1. Let p 2 be a prime, and let u F . The quadratic symbol modulo p of ≥ ∈ p a nonzero element u is defined by u 1 if u is a quadratic residue; = (5) p 1 if u is not a quadratic residue. (− In term of calculations, this can be determined via the Euler criterion u (p 1)/2 = u − 1mod p. (6) p ≡± Lemma 2.1. (Legendre) Let p 2 and q be a pair of distinct primes. Then, the quadratic symbol satisfies the following properties.≥ a (p 1)/2 (i) a − mod p, Euler congruence equation. p ≡ Consecutive Quadratic Residues And Quadratic Nonresidues Modulo p 3 ab a b (ii) = , completely multiplicative function. p p p 1 (p 1)/2 (iii) − = ( 1) − , evaluation at a = 1. p − − 2 (p2 1)/8 (iv) = ( 1) − , evaluation at a = 2. p − q p (p 1)(q 1)/4 (v) = ( 1) − − , quadratic reciprocity. p q − 3 Some Quadratic Exponential Sums Definition 3.1. Let p 2 be a prime. The finite Fourier transform of a periodic function ≥ f : Z C of period p is defined by −→ 1 fˆ(s)= f(t)ei2πst/p, (7) ηp√p t Fp X∈ where η =1 if p 1mod4 or η = i if p 3 mod 4. p ≡ p ≡ Except for a normalizing factor, the standard Gauss sum is a finite Fourier transform of the nonprincipal character χ : Z C, namely, −→ i2πts/p τs(χ)= χ(t)e . (8) t Fp X∈ Lemma 3.1. The quadratic character mod p is the unique fix point of the finite Fourier transform. Specifically s 1 t = ei2πst/p, (9) p ηp√p p t Fp X∈ where η = 1 if p 1mod4 or η = i if p 3mod4. p ≡ p ≡ Lemma 3.2. (Gauss) If p 2 is a prime, then ≥ 2 √p if p 1mod p; ei2πu /p = ≡ (10) i√p if p 3mod p. u Fp ( X∈ ≡ Proof. It is widely available in the literature, exampli gratia, [7, Theorem 1.1.5], [15, Section 3.3], [20, Lemma 3.3]. Lemma 3.3. Let p 2 be a prime, and let (x p) be the quadratic character mod p. If ≥ | the element s = 0, then, 6 1 2 s ei2πu s/p = − η √p, (11) p p u Fp X∈ where η = 1 if p 1mod4 or η = i if p 3mod4. p ≡ p ≡ Consecutive Quadratic Residues And Quadratic Nonresidues Modulo p 4 Proof. Let χ(n) = (x p). Replace the characteristic function of quadratic residue, see | Lemma 5.1, to obtain 2 ei2πu s/p = (1 + χ(u)) ei2πus/p = χ(u)ei2πus/p. (12) u Fp u Fp u Fp X∈ X∈ X∈ The change of variable z = us returns i2πu2s/p 1 i2πz/p 1 e = χ(s− ) χ(z)e = χ(s− )ηp√p. (13) u Fp z Fp X∈ X∈ Lemma 3.4. If p 2 is a prime, and a = is an integer such that gcd(a, p) = 1, then ≥ 6 a 2 ax2 + bx + c if b 4ac 0mod p; = − p − 6≡ (14) p a (p 1) if b2 4ac 0mod p. u Fp p X∈ − − ≡ Proof. Consult the literature, [7, Theorem 2.1.2], [17], and similar references. 4 Some Incomplete Exponential Sums A classical application of the finite Fourier transform provides nontrivial upper bounds of incomplete character sums. The simplest one uses the quadratic symbol or plain character χ modulo q. Lemma 4.1. (Polya-Vinogradov) For q is a large prime, and a character χ = 1 modulo 6 q, χ(n) √q log q. (15) ≪ n x X≤ Proof. Use the finite Fourier transform of χ(n) as in Lemma 3.1 and the geometric series, and other means. The distribution, and various properties of the implied constant has a vast literature, and it is a topic of current research, see [5], [12], et alii. Many improved upper bounds for some specific characters such as χ( 1) = 1 or χ( 1) = 1 are known. An explicit for the − − − Burgess inequality is stated below. Lemma 4.2. ([30]) Let p 107 be a prime, and let χ be character modulo p. Let M, and ≥ N 1 be nonnegative integer and let r 1. Then ≥ ≥ 1 1/r (r+1)/4r2 1/r χ(n) 2.7N − p (log p) . (16) ≤ M n N+M ≤ X≤ At r = 1 it reduces to the Polya-Vinogradov inequality, and as r , it becomes a trivial →∞ upper bound. Lemma 4.3. Suppose that GRH is true. Then, for any nonprincipal character χ modulo q and any large number x, χ(n) √q log log q. (17) ≪ n x X≤ Consecutive Quadratic Residues And Quadratic Nonresidues Modulo p 5 Proof. This is done in [22, Theorem 2]. Lemma 4.4. (Paley) There are infinitely many discriminant q 1mod q for which ≡ n √q log log q. (18) q ≫ n x X≤ Proof. The original version appears in [26] and a recent version is given in [21, Theorem 9.24]. 5 Characteristic Functions For Quadratic Residues The standard characteristic function of quadratic residues and quadratic nonresidues are induced by the quadratic symbol. Lemma 5.1. Let p 2 be a prime, and let (x p) be the quadratic character mod p. If ≥ | u F is a nonzero element, then, ∈ p (p 1)/2 1 u 1 if u − 1mod p, (i) Ψ2 (u)= 1+ = ≡ 2 p 0 if u(p 1)/2 1mod p. − ≡− (p 1)/2 1 u 1 if u − 1mod p, (ii) Ψ2(u)= 1 = ≡− 2 − p 0 if u(p 1)/2 1mod p, − ≡ are the characteristic functions for quadratic residues and quadratic non residues modulo p respectively in the finite field Fp.
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