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Congruences for Wolstenholme Primes

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Congruences for Wolstenholme Primes Czechoslovak Mathematical Journal, 65 (140) (2015), 237–253 CONGRUENCES FOR WOLSTENHOLME PRIMES Romeo Meštrovic´, Kotor (Received January 21, 2014) Abstract. A prime p is said to be a Wolstenholme prime if it satisfies the congruence 2p−1 ≡ 4 2p−1 8 p−1 1 (mod p ). For such a prime p, we establish an expression for p−1 (mod p ) p−1 i given in terms of the sums Ri := 1/k (i = 1, 2, 3, 4, 5, 6). Further, the expression in =1 kP this congruence is reduced in terms of the sums Ri (i = 1, 3, 4, 5). Using this congruence, we prove that for any Wolstenholme prime p we have p−1 p−1 2p − 1 1 2 1 7 ≡ 1 − 2p − 2p (mod p ). p − 1 k k2 kX=1 kX=1 Moreover, using a recent result of the author, we prove that a prime p satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique of Helou and Terjanian, the above congruence is given as an expression involving the Bernoulli numbers. Keywords: congruence; prime power; Wolstenholme prime; Wolstenholme’s theorem; Bernoulli number MSC 2010 : 11B75, 11A07, 11B65, 11B68, 05A10 1. Introduction and statements of results Wolstenholme’s theorem (see, e.g., [23], [7]) asserts that if p is a prime greater 2p−1 than 3, then the binomial coefficient p−1 satisfies the congruence 2p − 1 ≡ 1 (mod p3). p − 1 It is well known (see, e.g., [8]) that this theorem is equivalent to the assertion that for any prime p > 5 the numerator of the fraction 1 1 1 1+ + + ... + , 2 3 p − 1 237 written in reduced form, is divisible by p2. A. Granwille [7] established broader generalizations of Wolstenholme’s theorem. As an application, it is obtained in [7] that for a prime p > 5 we have 2p − 1 2p 3 3 2 3 ≡ (mod p5). p − 1 . p 2.1 Notice that C. Helou and G. Terjanian [9] established many Wolstenholme type con- gruences modulo pk with a prime p and k ∈ N such that k 6 6. One of their np main results ([9], Proposition 2, pages 488–489) is a congruence of the form mp ≡ n > N 6 6 f(n,m,p) m (mod p), where p 3 is a prime number, m,n ∈ with 0 m n, and f is a function on m,n and p involving the Bernoulli numbers Bk. As an ap- plication, ([9], Corollary 2 (2), page 493; also see Corollary 6 (2), page 495), for any prime p > 5 we have 2p − 1 3 1 5 6 5 6 ≡ 1 − p B 3 2 + p Bp − p Bp (mod p ). p − 1 p −p −2 3 −3 5 −5 A similar congruence modulo p7 (Corollary 1.2) is obtained in this paper for Wol- stenholme primes. A prime p is said to be a Wolstenholme prime if it satisfies the congruence 2p − 1 ≡ 1 (mod p4). p − 1 The two known such primes are 16843 and 2124679, and in 2007 R. J. McIntosh and E. L. Roettger [17] reported that these primes are the only two Wolstenholme primes less than 109. However, using an argument based on the prime number theorem, McIntosh [16], page 387, conjectured that there are infinitely many Wolstenholme 2p−1 5 primes, and that no prime satisfies the congruence p−1 ≡ 1 (mod p ). Wolstenholme primes form a subset of irregular primes. Indeed, Wolstenholme primes are those irregular primes p which divide the numerator of Bp−3 (see, e.g., [16] or [19]). Recall that the irregular primes as well as Wieferich and related primes are connected with the first case of Fermat’s last theorem; see [21], Lecture I, pages 9–12, and [21], Lecture VIII, pages 151–154, [2], [3], [12], [13], [22]. The following result is basic in our investigations. Proposition 1.1. Let p be a Wolstenholme prime. Then p−1 p−1 p−1 p−1 2p − 1 1 p2 1 p3 1 p4 1 ≡ 1+ p − + − p − 1 k 2 k2 3 k3 4 k4 Xk=1 Xk=1 kX=1 kX=1 p−1 p−1 p5 1 p6 1 + − (mod p8). 5 k5 6 k6 Xk=1 Xk=1 238 The above congruence can be simplified as follows. Proposition 1.2. Let p be a Wolstenholme prime. Then p−1 p−1 p−1 p−1 2p − 1 3p 1 p2 1 7p3 1 5p5 1 ≡ 1+ − + + (mod p8). p − 1 2 k 4 k2 12 k3 12 k5 kX=1 Xk=1 kX=1 Xk=1 Reducing the modulus in the previous congruence, we can obtain the following simpler congruences. Corollary 1.1. Let p be a Wolstenholme prime. Then p−1 p−1 2p − 1 1 1 ≡ 1 − 2p − 2p2 p − 1 k k2 Xk=1 kX=1 p−1 p−1 1 2p3 1 ≡ 1+2p + (mod p7). k 3 k3 Xk=1 kX=1 The Bernoulli numbers Bk (k ∈ N) are defined by the generating function ∞ xk x B = . k k! ex − 1 Xk=0 It is easy to find the values B0 =1, B1 = −1/2, B2 =1/6, B4 = −1/30, and Bn =0 n−1 for odd n > 3. Furthermore, (−1) B2n > 0 for all n > 1. These and many other properties can be found, for instance, in [10] or [4]. The second congruence from Corollary 1.1 can be given in terms of the Bernoulli numbers by the following result. Corollary 1.2. Let p be a Wolstenholme prime. Then 2p − 1 3 3 5 3 6 7 ≡ 1 − p B 4 3 − p B 2 + p Bp (mod p ). p − 1 p −p −2 2 p −p−4 10 −5 The above congruence can be given by the following expression involving lower order Bernoulli numbers. 239 Corollary 1.3. Let p be a Wolstenholme prime. Then 2p − 1 3 8 8 1 ≡ 1 − p Bp−3 − 3B2p−4 + B3p−5 − B4p−6 p − 1 3 5 3 4 8 3 24 2 − p Bp−3 − B2p−4 + B3p−5 − B4p−6 9 2 25 9 5 8 3 72 4 12 − p Bp−3 − B2p−4 + B3p−5 − B4p−6 + Bp−5 − B2p−6 27 4 125 27 5 2 − p6B (mod p7). 25 p−5 Combining the first congruence in Corollary 1.1 and a recent result of the author in [18], Theorem 1.1, we obtain a new characterization of Wolstenholme primes as follows. Corollary 1.4 ([18], Remark 1.6). A prime p is a Wolstenholme prime if and only if p−1 p−1 2p − 1 1 1 ≡ 1 − 2p − 2p2 (mod p7). p − 1 k k2 Xk=1 kX=1 Remark 1.1. A computation shows that no prime p < 105 satisfies the second congruence in Corollary 1.1, except the Wolstenholme prime 16843. Accordingly, an interesting question is as follows: Is it true that the second congruence in Corol- lary 1.1 implies that a prime p is necessarily a Wolstenholme prime? We conjecture that this is true. A proof of Proposition 1.1 is given in the next section. Proofs of Proposition 1.2 and Corollaries 1.1–1.3 are presented in Section 3. 2. Proof of Proposition 1.1 For the proof of Proposition 1.1, we will need some auxiliary results. Lemma 2.1. For any prime p > 7, we have p−1 p−1 1 1 (2.1) 2 ≡−p (mod p4). k k2 Xk=1 Xk=1 Proof. The above congruence is in fact the congruence (14) in ([25], Proof of Theorem 3.2). 240 Lemma 2.2. For any prime p > 7, we have p−1 2p − 1 1 (2.2) ≡ 1+2p (mod p5), p − 1 k Xk=1 and p−1 2p − 1 1 (2.3) ≡ 1 − p2 (mod p5). p − 1 k2 Xk=1 p−1 2 Proof. Let R1(p)= 1/k. Following ([25], Definition 3.1) we define wp <p k=1 P 2 2 to be the unique nonnegative integer such that wp ≡ R1(p)/p (mod p ). Then by ([25], Theorem 3.2), for all nonnegative integers n and r with n > r, np n 3 5 (2.4) ≡ 1+ wpnr(n − r)p (mod p ). rpr 1 2p 2p−1 Since 2 p = p−1 , taking n =2 and r =1, (2.4) becomes 2p − 1 ≡ 1+2w p3 (mod p5), p − 1 p which is actually (2.2). Now the congruence (2.3) follows immediately from (2.2) and (2.1) of Lemma 2.1. Lemma 2.3. The following statements about a prime p > 7 are equivalent: (i) p is a Wolstenholme prime; p−1 (ii) 1/k ≡ 0 (mod p3); k=1 pP−1 (iii) 1/k2 ≡ 0 (mod p2); kP=1 (iv) p divides the numerator of the Bernoulli number Bp−3. P r o o f. The equivalences (i)⇔(ii)⇔(iii) are immediate from Lemma 2.2 if we consider the congruences (2.2) and (2.3) modulo p4. Further, by a special case of Glaisher’s congruence ([5], page 21, [6], page 323; also see [16], Theorem 2), we have 2p − 1 2 3 4 ≡ 1 − p Bp (mod p ), p − 1 3 −3 which implies the equivalence (i)⇔(iv). This concludes the proof. 241 For the proof of Proposition 1.1, we use the congruences (2.2) and (2.3) of Lemma 2.2 with (mod p4) instead of (mod p5). By a classical result of E.Leh- p−1 1 3 mer [15]; (also see [24], Theorem 2.8), 1/k ≡ − 3 Bp−3 (mod p ).
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