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Results for Nonwieferich Primes City University of New York (CUNY) CUNY Academic Works Publications and Research Bronx Community College 2019 Results for nonWieferich Primes Nelson A. Carella CUNY Bronx Community College How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/bx_pubs/64 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Results for nonWieferich Primes N. A. Carella Abstract: Let v ≥ 2 be a fixed integer, and let x ≥ 1 be a large number. The exact asymptotic counting function for the number of nonWieferich primes p ≤ x such that vp−1 − 1 ≡ 0 mod p2 in the interval [1; x] is proposed in this note. The current results in the literature provide lower bounds, which are conditional on the abc conjecture or the Erdos binary additive conjecture. 1 Introduction Let P = f2; 3; 5; 7;:::g be the set of prime numbers, and let v ≥ 2 be a fixed integer. The subsets of Wieferich primes and nonWieferich primes are defined by p−1 2 Wv = fp 2 P : v − 1 ≡ 0 mod p g (1) and p−1 2 Wv = fp 2 P : v − 1 6≡ 0 mod p g respectively. The set of primes P = Wv [ Wv is a disjoint union of these subsets. The subsets Wv and Wv have other descriptions by means of the orders ordp(v) = d j p − 1 and × 2 × ordp2 (v) = pd 6= p of the base v in the cyclic group (Z=pZ) and Z=p Z respectively. The order m−1 is defined by ordn(v) = minfm ≥ 1 : v − 1 ≡ 0 mod ng. Specifically, Wv = p : ordp2 (v) j p − 1 (2) and Wv = p : ordp2 (v) - p − 1 : For a large number x ≥ 1, the corresponding counting functions for the number of such primes up to x are defined by Wv(x) = # p ≤ x : ordp2 (v) j p − 1 (3) and W v(x) = π(x) − Wv(x); where π(x) = #fp ≤ xg is the primes counting function, respectively. Assuming the abc conjecture, several authors have proved that there are infinitely many non- Wieferich primes, see [13], [4], et alii. These results have lower bounds of the form log x W (x) (4) v log log x or slightly better. In addition, assuming the Erdos binary additive conjecture, there is a proof that the subset of nonWieferich primes has nonzero density. More precisely, x W (x) ≥ c (5) v log x July 29, 2019 AMS MSC :Primary 11A41; Secondary 11B25. Keywords: Distribution of primes; Wieferich Prime; nonWieferich Prime 1 Results for nonWieferich Primes 2 where c > 0 is a constant, see [5, Theorem 1] for the details. Here, it is shown that the subset of nonWieferich primes has density 1 in the set of primes unconditionally. Theorem 1.1. Let v ≥ 2 be a small base, and let x ≥ 1 be a large number. Then, the number of nonWieferich primes has the asymptotic formula x x W v(x) = + O : (6) log x log2 x Proof. The upper bound Wv(x) = O (log log x), confer Theorem 6.1, is used below to derive a lower bound for the counting function W v(x). This is as follows: W v(x) = # p ≤ x : ordp2 (v) - p − 1 = π(x) − Wv(x) (7) ≥ π(x) + O (log log x) x x = + O ; log x log2 x 2 where π(x) = #fp ≤ xg = x=log x + O(x=log x). Corollary 1.1. Almost every odd integer is a sum of a squarefree number and a power of two. Proof. Same as Theorem 1 in [5], but use the sharper upper bound Wv(x) = O (log log x). 2 Characteristic Functions Modulo Prime Powers The standard method for constructing characteristic function for primitive elements are discussed in [11, Corollary 3.5], [9, p. 258], and some characteristic functions for finite rings are discussed 2 × in [6]. These type of characteristic functions detect the orders of the elements v 2 Z=p Z by means of the divisors of '(p2). A new method for constructing characteristic functions in cyclic groups is developed here. These 2 × type of characteristic functions detect the orders of the elements v 2 Z=p Z by means of the solutions of the equation τ pn − v ≡ 0 mod p2, where v; τ are constants, and n is a variable such Q that 1 ≤ n < p − 1, and gcd(n; p − 1) = 1. The formula '(n) = pjn(1 − 1=p) denotes the Euler totient function. 2 2 × Lemma 2.1. Let p ≥ 3 be a prime, and let τ be a primitive root mod p . Let v 2 Z=p Z be a nonzero element. Then i2π(τpn−v)m 1 1 if ord 2 (v) = p − 1; X X '(p2) p 2 e = '(p ) 0 if ordp2 (v) 6= p − 1: 1≤n<p−1 0≤m<'(p2) gcd(n;p−1)=1 2 × 2 Proof. Let τ 2 Z=p Z be a fixed primitive root of order p(p − 1) = '(p ). As the index n ≥ 1 pn 2 × ranges over the integers relatively prime to p − 1, the element τ 2 Z=p Z ranges over the pn elements of order ordp2 (τ ) = p − 1. Hence, the equation τ pn − v = 0 (8) 2 × has a solution if and only if the fixed element Z=p Z is an elements of order ordp2 (v) = p − 1. pn 2 Setting w = ei2π(τ −v)='(p ) and summing the inner sum yield X 1 X m 1 if ordp2 (v) = p − 1; 2 w = (9) '(p ) 0 if ordp2 (v) 6= p − 1: gcd(n;p−1)=1 0≤m<'(p2) P m x This follows from the geometric series identity 0≤m≤x−1 w = (w − 1)=(w − 1); w 6= 1 applied to the inner sum. Results for nonWieferich Primes 3 The characteristic function for any element v ≥ 2 of any order ordp2 (v) = d j p − 1 in the cyclic 2 × group Z=p Z is a sum of characteristic functions. Lemma 2.2. Let Let v ≥ 2 be a fixed base, let p ≥ 3 be a prime, and let τ be a primitive root mod p2. The indicator function for the subset of primes such that vp−1 − 1 ≡ 0 mod p2 is given by X X 1 X i2π(τdpn−v)m Ψ (p2) = e '(p2) 0 '(p2) djp−1 1≤n<p−1 0≤m<'(p2) gcd(n;(p−1)=d)=1 1 if ord 2 (v) j p − 1; = p 0 if ordp2 (v) 6j p − 1: dpn 2 Proof. Suppose that ordp2 (v) j p − 1. Then, the congruence τ − v ≡ 0 mod p has a unique solution pair d j p − 1 and n ≥ 1 with gcd(n; (p − 1)=d) = 1. Otherwise, τ dpn − v 6≡ 0 mod p2 for all pairs d j p − 1 and gcd(n; (p − 1)=d) = 1. To complete the proof, proceed as in the proof of Lemma 2.1. 3 Equivalent Exponential Sums For any fixed 0 6= s 2 Z=p2Z, an asymptotic relation for the exponential sums X n 2 X n 2 ei2πsτ ='(p ) and ei2πτ ='(p ); (10) gcd(n;'(p2))=1 gcd(n;'(p2))=1 is provided in Lemma 3.1. This result expresses the first exponential sum in (10) as a sum of simpler exponential sum and an error term. The proof is based on Lagrange resolvent in the finite ring Z=p2Z. Specifically, dp dp 2dp (p−1)dp (!t; ζsτ ) = ζs + !−tζsτ + !−2tζsτ + ··· + !−(p−1)tζsτ ; (11) 2 where ! = ei2π=p, ζ = ei2π='(p ), and the variables 0 6= s 2 Z=p2Z, and 0 6= t 2 Z=pZ. Lemma 3.1. Let p ≥ 2 be a large prime. If τ be a primitive root modulo p2, then, X ndp 2 X ndp 2 ei2πsτ ='(p ) = ei2πτ ='(p ) + O(p1=2 log3 p); gcd(n;(p−1)=d)=1 gcd(n;(p−1)=d)=1 for any fixed d j p − 1, and 0 6= s 2 Z=p2Z. Proof. Summing (11) times !tn over the variable t 2 Z=pZ yields, (all the nontrivial complex pth root of unity), ndp 2 X ndp p · ei2πsτ ='(p ) = (!t; ζsτ )!tn: (12) 0≤t≤p−1 Summing (12) over the variable n ≥ 1, for which gcd(n; (p − 1)=d) = 1, yields X ndp 2 X X dp p · ei2πsτ ='(p ) = (!t; ζsτ )!tn gcd(n;(p−1)=d)=1 gcd(n;(p−1)=d)=1; 0≤t≤p−1 X dp X = (!t; ζsτ ) !tn − p: (13) 1≤t≤p−1 gcd(n;(p−1)=d)=1 The first index t = 0 contributes p, see [10, Equation (5)] for similar calculations. Likewise, the basic exponential sum for s = 1 can be written as X ndp 2 X dp X p · ei2πτ ='(p ) = (!t; ζτ ) !tn − p; (14) gcd(n;(p−1)=d)=1 1≤t≤p−1 gcd(n;(p−1)=d)=1 Results for nonWieferich Primes 4 Differencing (13) and (14) produces 0 1 X i2πsτ ndp='(p2) X i2πτ ndp='(p2) S1 = p · @ e − e A gcd(n;(p−1)=d)=1 gcd(n;(p−1)=d)=1 X dp dp X = (!t; ζsτ ) − (!t; ζτ ) !tn: (15) 1≤t≤p−1 gcd(n;(p−1)=d)=1 The right side sum S1 can be rewritten as X t sτ ndp t τ ndp X tn S1 = (! ; ζ ) − (! ; ζ ) ! 1≤t≤p−1 gcd(n;(p−1)=d)=1 et et( p−1 +1) X ndp ndp X ! − ! d = (!t; ζsτ ) − (!t; ζτ ) µ(e) (16) 1 − !et 1≤t≤p−1 e≤(p−1)=d et et( p−1 +1) X X ndp ndp ! − ! d = (!t; ζsτ ) − (!t; ζτ ) µ(e) : 1 − !et 1≤t≤p−1; e≤(p−1)=d The second line follows from Lemma 3.2-i.
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