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Results for nonWieferich Primes
Nelson A. Carella CUNY Bronx Community College
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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 = {2, 3, 5, 7,...} 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 = {p ∈ P : v − 1 ≡ 0 mod p } (1) and p−1 2 Wv = {p ∈ P : v − 1 6≡ 0 mod p } 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 | 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) = min{m ≥ 1 : v − 1 ≡ 0 mod n}. Specifically, Wv = p : ordp2 (v) | 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) | p − 1 (3) and W v(x) = π(x) − Wv(x), where π(x) = #{p ≤ x} 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) = #{p ≤ x} = 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 ∈ Z/p Z by means of the divisors of ϕ(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 ∈ 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 ϕ(n) = p|n(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 ∈ Z/p Z be a nonzero element. Then
i2π(τpn−v)m 1 1 if ord 2 (v) = p − 1, X X ϕ(p2) p 2 e = ϕ(p ) 0 if ordp2 (v) 6= p − 1. 1≤n 2 × 2 Proof. Let τ ∈ Z/p Z be a fixed primitive root of order p(p − 1) = ϕ(p ). As the index n ≥ 1 pn 2 × ranges over the integers relatively prime to p − 1, the element τ ∈ 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)/ϕ(p ) and summing the inner sum yield X 1 X m 1 if ordp2 (v) = p − 1, 2 w = (9) ϕ(p ) 0 if ordp2 (v) 6= p − 1. gcd(n,p−1)=1 0≤m<ϕ(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 | 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 ϕ(p2) 0 ϕ(p2) d|p−1 1≤n dpn 2 Proof. Suppose that ordp2 (v) | p − 1. Then, the congruence τ − v ≡ 0 mod p has a unique solution pair d | p − 1 and n ≥ 1 with gcd(n, (p − 1)/d) = 1. Otherwise, τ dpn − v 6≡ 0 mod p2 for all pairs d | 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 ∈ Z/p2Z, an asymptotic relation for the exponential sums X n 2 X n 2 ei2πsτ /ϕ(p ) and ei2πτ /ϕ(p ), (10) gcd(n,ϕ(p2))=1 gcd(n,ϕ(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 (ωt, ζsτ ) = ζs + ω−tζsτ + ω−2tζsτ + ··· + ω−(p−1)tζsτ , (11) 2 where ω = ei2π/p, ζ = ei2π/ϕ(p ), and the variables 0 6= s ∈ Z/p2Z, and 0 6= t ∈ 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 /ϕ(p ) = ei2πτ /ϕ(p ) + O(p1/2 log3 p), gcd(n,(p−1)/d)=1 gcd(n,(p−1)/d)=1 for any fixed d | p − 1, and 0 6= s ∈ Z/p2Z. Proof. Summing (11) times ωtn over the variable t ∈ Z/pZ yields, (all the nontrivial complex pth root of unity), ndp 2 X ndp p · ei2πsτ /ϕ(p ) = (ωt, ζsτ )ωtn. (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 /ϕ(p ) = (ωt, ζsτ )ωtn gcd(n,(p−1)/d)=1 gcd(n,(p−1)/d)=1, 0≤t≤p−1 X dp X = (ωt, ζsτ ) ωtn − 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 /ϕ(p ) = (ωt, ζτ ) ωtn − 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 X i2πsτ ndp/ϕ(p2) X i2πτ ndp/ϕ(p2) S1 = p · e − e gcd(n,(p−1)/d)=1 gcd(n,(p−1)/d)=1 X dp dp X = (ωt, ζsτ ) − (ωt, ζτ ) ωtn. (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 = (ω , ζ ) − (ω , ζ ) ω 1≤t≤p−1 gcd(n,(p−1)/d)=1 et et( p−1 +1) X ndp ndp X ω − ω d = (ωt, ζsτ ) − (ωt, ζτ ) µ(e) (16) 1 − ωet 1≤t≤p−1 e≤(p−1)/d et et( p−1 +1) X X ndp ndp ω − ω d = (ωt, ζsτ ) − (ωt, ζτ ) µ(e) . 1 − ωet 1≤t≤p−1, e≤(p−1)/d The second line follows from Lemma 3.2-i. The upper bound et et( p−1 +1) X X ndp ndp ω − ω d |S | ≤ (ωt, ζsτ ) − (ωt, ζτ ) µ(e) 1 1 − ωet 1≤t≤p−1, e≤(p−1)/d et et( p−1 +1) X X ndp ndp ω − ω d ≤ (ωt, ζsτ ) − (ωt, ζτ ) µ(e) 1 − ωet 1≤t≤p−1, e≤(p−1)/d et et( p−1 +1) X X ω − ω d ≤ 2p1/2 log p · µ(e) 1 − ωet 1≤t≤p−1, e≤(p−1)/d X 2p log p ≤ 2p1/2 log p · πt 1≤t≤p−1 X 1 ≤ 4p3/2 log2 p (17) t 1≤t≤p−1 ≤ 8p3/2 log3 p. The third line follows the upper bound for Lagrange resolvents, and the fourth line follows from Lemma 3.2-ii. Here, the difference of two Lagrange resolvents, (Gauss sums), has the upper bound t sτ dp t τ dp X i2πt/p 1/2 (ω , ζ ) − (ω , ζ ) ≤ 2 χ(t)e ≤ 2p log p, (18) 1≤t≤p−1 where |χ(t)| = 1 is a root of unity. Taking absolute value in (15) and using (17) return X i2πsτ ndp/p X i2πτ ndp/p p · e − e ≤ |S1| gcd(n,(p−1)/d)=1 gcd(n,(p−1)/d)=1 ≤ 8p3/2 log3 p. (19) The last inequality implies the claim. Lemma 3.2. Let p ≥ 2 be a large prime, and let ω = ei2π/p be a pth root of unity. Then, (i) et et( p−1 +1) X X ω − ω d ωtn = µ(e) , 1 − ωet gcd(n,(p−1)/d)=1 e≤(p−1)/d Results for nonWieferich Primes 5 (ii) X tn 2p log p ω ≤ , πt gcd(n,(p−1)/d)=1 where µ(k) is the Mobius function, for any fixed pair d | p − 1 and t ∈ [1, p − 1]. Proof. (i) Use the inclusion exclusion principle to rewrite the exponential sum as X X X ωtn = ωtn µ(e) gcd(n,(p−1)/d)=1 n≤(p−1)/d e|(p−1)/d e|n X X = µ(e) ωtn e≤(p−1)/d n≤(p−1)/d e|n X X = µ(e) ωetm (20) e≤(p−1)/d m≤(p−1)/de et et( p−1 +1) X ω − ω d = µ(e) . 1 − ωet e≤(p−1)/d (ii) Observe that the parameters ω = ei2π/p, the integers t ∈ [1, p − 1], and e ≤ (p − 1)/d imply that πet/p 6= kπ with k ∈ Z, so the sine function sin(πet/p) 6= 0 is well defined. Using standard manipulations, and z/2 ≤ sin(z) < z for 0 < |z|< π/2, the last expression becomes et et( p−1 +1) ω − ω d 2 2p ≤ ≤ (21) et 1 − ω sin(πet/p) πet for 1 ≤ d ≤ p − 1. Finally, the upper bound is et et( p−1 +1) X ω − ω d 2p X 1 µ(e) ≤ (22) 1 − ωet πt e e≤(p−1)/d e≤(p−1)/d 2p log p ≤ . πt 4 Upper Bound For The Main Term An estimate for the finite sum occurring in the evaluation of the main term is considered in this section. Lemma 4.1. Let x ≥ 1 be a large number, and let ϕ(n) be the Euler totient function. Then X 1 X X 1 ≤ 2 log log x. ϕ(p2) p≤x d|p−1, gcd(n,(p−1)/d)=1 P Proof. Use the identity d|n ϕ(d) = n to eliminate the inner double sum in the following way: X X X 1 = ϕ((p − 1)/d) = p − 1. (23) d|p−1, gcd(n,(p−1)/d)=1 d|p−1 Substituting this returns X 1 X X X 1 X 1 1 = · (p − 1) = . (24) ϕ(p2) ϕ(p2) p p≤x d|p−1, gcd(n,(p−1)/d)=1 p≤x p≤x Lastly, apply Mertens theorem to the prime harmonic sum. Results for nonWieferich Primes 6 5 Estimate For The Error Term Upper bound for the error term occurring in the proof of Theorem 1.1 is determined here. This estimate is weak, but sufficient in many applications. 2 × Lemma 5.1. Let x ≥ 1 be large number. Let p ≥ 2 be a large prime, and let τ ∈ Z/p Z be a primitive root mod p2. If the element v ≥ 2 is a small fixed integer, then X X X 1 X i2π(τpdn−v)m e ϕ(p2) = O (log log x) , ϕ(p2) p≤x, d|p−1, gcd(n,(p−1)/d)=1 1≤m<ϕ(p2) for all sufficiently large numbers x ≥ 1. Proof. Rearrange the inner triple finite sum in the form X X X 1 X i2π(τdpn−v)m T = e ϕ(p2) (25) ϕ(p2) p≤x d|p−1, gcd(n,(p−1)/d)=1 1≤m<ϕ(p2) dpn X 1 X −i2π vm X X i2π mτ = e ϕ(p2) e ϕ(p2) ϕ(p2) p≤x 0 The last line in (25) follows from Lemma 3.1. Taking absolute the value of the first term T1 yields X −i2πvm/ϕ(p2) |T1| = e = w ≤ v (26) 0 d|p−1 gcd(n,(p−1)/d)=1 X X 1 + p1/2 log4 p d|p−1 gcd(n,(p−1)/d)=1 X p − 1 ϕ + p1/2 log4 p d d|p−1 p − 1. (27) Replace the identity ϕ(p2) = p(p − 1), and the estimates (26) and (27) back into (25). These lead to X 1 |T | ≤ |T | × |T | (28) ϕ(p2) 1 2 p≤x X 1 (v × (p − 1)) (p − 1)p p≤x X 1 p p≤x log log x, where v ≥ 2 is the base. Results for nonWieferich Primes 7 6 Upper Bound For The Wieferich Primes The subset of primes W2 = p : ordp2 (2) | p − 1 = {1093, 3511,..., } associated with the base v = 2 is the best known case. But, many other bases have been computed too, see [2], [8]. The heuristic argument in [12, p. 413], [1], [7, Section 2], et alii, claims that X 1 W (x) ≈ log log x. (29) v p p≤x The basic foundation of this heuritic is correct. An unconditional upper bound is computed here. Theorem 6.1. The number of Wieferich primes on the interval [1, x] has the upper bound Wv(x) = O (log log x) . Proof. Let x ≥ 1 be a large number, and fix an integer v ≥ 2. The sum of the characteristic function over the interval [1, x] is written as X 2 Wv(x) = Ψ0(p ). (30) p≤x Replacing the characteristic function, see Lemma 2.2, and expanding yield X X X X 1 X i2π(τpn−v)m Ψ (p2) = e ϕ(p2) v ϕ(p2) p≤x p≤x, d|p−1, gcd(n,(p−1)/d)=1 0≤m<ϕ(p2) X 1 X X = 1 (31) ϕ(p2) p≤x d|p−1, gcd(n,(p−1)/d)=1 X X X 1 X i2π(τpn−v)m + e ϕ(p2) ϕ(p2) p≤x, d|p−1, gcd(n,(p−1)/d)=1 1≤m<ϕ(p2) = Mv(x) + Ev(x). The main term Mv(x) is determined by the index m = 0, and the error term Ev(x) is determined by the range 1 ≤ m < ϕ(p2). Applying Lemma 4.1 to the main term and applying Lemma 5.1 to the error term yield Wv(x) = Mv(x) + Ev(x) ≤ 2 log log(x) + O (log log x) = O (log log x) . (32) This verifies the upper bound. References [1] Richard Crandall, Karl Dilcher, Carl Pomerance. A search for Wieferich and Wilson primes. Math. Comp. 66 (1997), no. 217, 433–449. [2] Francois GDorais, Dominic Klyve. A Wieferich prime search up to 6.7 × 1015. J. Integer Seq. 14 (2011), no. 9, Article 11.9.2, 14 pp. [3] J.-M. DeKoninck, N. Doyon . On the set of Wieferich primes and of its complement. 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