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On a Congruence Modulo N Involving Two Consecutive Sums of Powers

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On a Congruence Modulo N Involving Two Consecutive Sums of Powers 1 2 Journal of Integer Sequences, Vol. 17 (2014), 3 Article 14.8.4 47 6 23 11 On a Congruence Modulo n3 Involving Two Consecutive Sums of Powers Romeo Meˇstrovi´c Maritime Faculty University of Montenegro 85330 Kotor Montenegro [email protected] Abstract For various positive integers k, the sums of kth powers of the first n positive integers, k k k Sk(n):=1 +2 + +n , are some of the most popular sums in all of mathematics. In ··· 3 this note we prove a congruence modulo n involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n 5 is a prime such that n 1 ∤ 2k 2, then ≥ − − this congruence is satisfied modulo n4. This suggests a conjecture about when a prime can be a Wolstenholme prime. We also propose several Giuga-Agoh-like conjectures. Further, we establish two congruences modulo n3 for two binomial-type sums involving sums of powers S2i(n) with i = 0, 1,...,k. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums. 1 Introduction and basic results n k The sum of powers of integers i=1 i is a well-studied problem in mathematics (see, e.g., [9, 40]). Finding formulas for these sums has interested mathematicians for more than 300 years since the time of Jakob BernoulliP (1654–1705). Several methods were used to find the sum Sk(n) (see, for example, Vakil [49]). These lead to numerous recurrence relations. For a nice account of sums of powers, see Edwards [15]. For simplicity, here as always in the 1 sequel, for all integers k 1 and n 2 we write ≥ ≥ n−1 S (n) := ik =1k +2k +3k + +(n 1)k. k ··· − i=1 X The study of these sums led Jakob Bernoulli to develop numbers later named in his honor. Namely, the celebrated Bernoulli formula (sometimes called Faulhaber’s formula) gives the sum Sk(n) explicitly as (see, e.g., Beardon [4]) 1 k k +1 S (n)= nk+1−iB , (1) k k +1 i i i=0 X where Bi (i =0, 1, 2,...) are the Bernoulli numbers defined by the generating function ∞ xi x B = . i i! ex 1 i=0 X − 1 1 1 It is easy to find the values B0 = 1, B1 = 2 , B2 = 6 , B4 = 30 , and Bi = 0 for odd i 3. i−1 − − ≥ Furthermore, ( 1) B2i > 0 for all i 1. These and many other properties can be found, for instance, in− [23]. Several generalizations≥ of the formula (1) were established by Z.-H. Sun ([46, Thm. 2.1] and [47]) and Z.-W. Sun [48]. By the well-known Pascal’s identity proven by Pascal in 1654 (see, e.g., [29]), we have k−1 k S (n +1)=(n + 1)k 1. (2) i i − i=0 X Recall also that the formula (2) is also presented in Bernoulli’s Ars Conjectandi [6], (also see [19, pp. 269–270]) published posthumously in 1713. On the other hand, divisibility properties of the sums Sk(n) were investigated by many authors [13, 27, 30, 42]. For example, in 2003 Damianou and Schumer [13, Thm. 1, p. 221 and Thm. 2, p. 222] proved, respectively: (1) if k is odd, then n divides Sk(n) if and only if n is incogruent to 2 modulo 4; (2) if k is even, then n divides Sk(n) if and only if n is not divisible by any prime p such that p D , where D is the denominator of the kth Bernoulli number B . | k k k Denominators of Bernoulli numbers Bk (k =0, 1, 2,...) are given as the sequence A027642 in [41] (cf. also its subsequence A002445 consisting of the terms with even indices k). Motivated by the recurrence formula for Sk(n) recently obtained in [34, Corollary 1.7], in this note we prove the following basic result. 2 Theorem 1. Let k and n be positive integers. Then for each k 2 ≥ 0 (mod n3), if k is even or n is odd 2S2k+1(n) (2k + 1)nS2k(n) or n 0 (mod 4); (3) − ≡ 3 ≡ n (mod n3), if k is odd and n 2 (mod 4). 2 ≡ Furthermore, 0 (mod n3), if n is odd; 2S3(n) 3nS2(n) n3 3 (4) − ≡ ( 2 (mod n ), if n is even. In particular, for all k 1 and n 1, we have ≥ ≥ 2S (n) (2k + 1)nS (n) (mod n2), (5) 2k+1 ≡ 2k and for all k 1 and n 2 (mod 4) ≥ 6≡ 2S (n) (2k + 1)nS (n) (mod n3). (6) 2k+1 ≡ 2k Combining the congruence (5) and the “even case” (2) of a result of Damianou and Schumer [13, Thm. 1, p. 221 and Thm. 2, p. 222] mentioned above, we obtain the following “odd” extension of their result. Corollary 2. If k is an odd positive integer and n a positive integer such that n is not divisible by any prime p such that p Dk−1, where Dk−1 is the denominator of the (k 1)th 2 | − Bernoulli number Bk−1, then n divides 2Sk(n). Conversely, if k is an odd positive integer and n a positive integer relatively prime to k 2 such that n divides 2Sk(n), then n is not divisible by any prime p such that p Dk−1, where D is the denominator of the (k 1)th Bernoulli number B . | k−1 − k−1 The paper is organized as follows. Some applications of Theorem 1 are presented in the following section. In Subsection 2.1 we give three particular cases of the congruence (3) of Theorem 1 (Corollary 3). One of these congruences immediately yields a reformulation of 2 3 Giuga’s conjecture in terms of the divisibility of 2Sn(n)+ n by n (Proposition 4). In the next subsection we establish the fact that the congruence (6) holds modulo n4 whenever n 5 is a prime such that n 1 ∤ 2k 2 ((14) of Proposition 7). Motivated by some particular≥ cases of this congruence− and related− computations in Mathematica 8, we propose several Giuga-Agoh-like conjectures. In particular, Conjecture 12 characterizes Wolstenholme primes as positive integers n such that S (n) 0 (mod n3). n−2 ≡ In Subsection 2.3 we establish two congruences modulo n3 for two binomial sums involving sums of powers S2i(n) with i =0, 1,...,k (Proposition 17). Combining the congruence (5) of Theorem 1 with the Carlitz-von Staudt result for de- termining S2k(n) (mod n) (Theorem 20), in the last subsection of Section 2, we extend this 2 result modulo n for power sums S2k+1(n) (Theorem 23). 3 Recall that Erd˝os-Moser Diophantine equation is the equation of the form 1k +2k + +(m 2)k +(m 1)k = mk (7) ··· − − where m 2 and k 2 are positive integers. Notice that (m, k)=(3, 1) is the only solution for k = 1.≥ In letter≥ to Leo Moser around 1950, Paul Erd˝os conjectured that such solutions of the above equation do not exist (see [39]). Using remarkably elementary methods, Moser [39] showed in 1953 that if (m, k) is a solution of (7) with m 2 and k 2, then m> 10106 ≥ ≥ and k is even. We believe that Theorem 23 can be useful for study some Erd˝os-Moser-like Diophantine equations with odd k (see Remark 26). Proofs of all our results are given in Section 3. 2 Applications of Theorem 1 2.1 Variations of Giuga-Agoh’s conjecture Taking k =(n 1)/2 if n is odd, k = n/2 if n is even and k =(n 2)/2 if n is even into (3) of Theorem 1 we− obtain respectively the following three congruences.− Corollary 3. If n is an odd positive integer, then 2S (n) n2S (n) (mod n3). (8) n ≡ n−1 If n is even, then 0 (mod n3), if n 0 (mod 4); 2Sn+1(n) n(n + 1)Sn(n) 3 ≡ (9) − ≡ n (mod n3), if n 2 (mod 4) ( 2 ≡ and 2S (n) n(n 1)S (n) (mod n3). (10) n−1 ≡ − n−2 In particular, for each even n we have 0 (mod n), if n 0 (mod 4); Sn−1(n) ≡ (11) ≡ n (mod n), if n 2 (mod 4). ( 2 ≡ Notice that if n is any prime, then by Fermat’s little theorem we have Sn−1(n) 1 (mod n). In 1950 Giuga [18] conjectured that a positive integer n 2 is a prime if and≡ only − ≥ if Sn−1(n) 1 (mod n). This conjecture is related to the sequences A029875, A007850, A198391, A199767≡ − , A226365 and A029876 in [41]. The following proposition provides an equivalent formulation of Giuga’s conjecture. Proposition 4. The following conjectures are equivalent: 4 (i) A positive integer n 3 is a prime if and only if ≥ S (n) 1 (mod n). (12) n−1 ≡− (ii) A positive integer n 3 is a prime if and only if ≥ 2S (n) n2 (mod n3). (13) n ≡− The above conjecture (ii) is related to the sequence A219540 in [41]. Since by the con- gruence (11), Sn−1(n) 1 (mod n) for each even n 4, without loss of generality Giuga’s conjecture may be restricted6≡ − to the set of odd positive≥ integers. In view of this fact and 2 the fact that by (8), n Sn(n) for each odd n, Proposition 4 yields the following equivalent formulation of Giuga’s conjecture.| Conjecture 5 (Giuga’s conjecture).
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