Perfect Power Riesel Numbers

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1 Perfect Power Riesel Numbers Carrie Finch a, Lenny Jones b a Mathematics Department, Washington and Lee University, Lexington, VA b Department of Mathematics, Shippensburg University, Shippensburg, PA Abstract A Riesel number k is an odd positive integer such that k 2 n 1 is composite for all integers n 1. In 2003, Chen proved that there are infinitely many Riesel numbers of the form k r, when r 0, 4, 6, 8 (mod 12), and he conjectured that such Riesel powers exist for all positive integers r. In 2008, Filaseta, Finch and Kozek extended Chen s theorem slightly by constructing Riesel numbers of the form k 4 and k 6. In this article, we extend these results further by proving that there exist infinitely many Riesel numbers of the form k r for any positive integer r that is coprime to 105. Keywords: Riesel number, covering 2010 MSC: 11B83, 11Y05 1. Introduction A Riesel number k is an odd positive integer with the property that k 2 n 1 is composite for all natural numbers n. The smallest known Riesel number is ; indeed, H. Riesel [9] showed that if k (mod ), then k is a Riesel number. It is believed that is the smallest Riesel number. As of this writing, there are 52 odd positive integers smaller that that are still candidates. See html for the most up-to-date information. In 2003, Y.G. Chen [2] showed that there are perfect power Riesel numbers for certain powers. In particular, for values of r that are either odd or twice an odd number not divisible by 3, he constructed integers k such that k r 2 n 1 is composite for all natural numbers n. Furthermore, Chen conjectured that there are Riesel numbers that are perfect rth powers for any positive integer r. In 2008, Filaseta, Finch and Kozek [5] provided further evidence for Chen s conjecture by proving for each n {4, 6} that there exists a set T n of Preprint submitted to Journal of Number Theory October 26, 2014

2 positive density such that each element in T n is a Riesel number of the form k r with r 0 (mod n). In this paper, we extend these previous results by establishing the following theorem. Theorem 1.1. For any positive integer r with gcd(r, 105) = 1, there exist infinitely many odd positive integers k such that k r 2 n 1 is composite for all integers n 1. Moreover, k r 2 n 1 has at least two distinct prime divisors for each value of n, when r Preliminaries The following concept, due to Erdős [4], is crucial to the proof of Theorem 1.1. Definition 2.1. A covering of the integers is a finite system of congruences x a i (mod m i ), where m i > 1, such that every integer n satisfies at least one of the congruences. For brevity of notation, we present a covering C as a set of ordered pairs (a i, m i ). We let L C denote the least common multiple of all the moduli m i occurring in C. Quite often when a covering C is used to solve a problem, there is a set of prime numbers associated with C. In the situation occurring in this article, for each (a i, m i ) C, there exists a corresponding prime p i, such that 2 m i 1 (mod p i ), where 2 s 1 (mod p i ) for all positive integers s < m i. We call such a prime a primitive divisor of 2 m i 1. In terms of group theory, a primitive divisor p of 2 m 1, where m > 1 is an integer, is a prime such that in the group of units modulo p, which we denote (Z p ), the element 2 has order m. We denote the order of the integer z modulo a prime p as ord p (z). A covering with certain restrictions on the moduli is used to establish Theorem 1.1. To build this covering, we can use a particular modulus m > 1 as many times as there are distinct primitive divisors of 2 m 1. It is well known that 2 m 1, with m > 1, has at least one primitive divisor as long as m 6. This result is originally due to Bang [1]. Two additional facts are needed here. The first result is due to Darmon and Granville [3]. Theorem 2.2 ([3]). Let A, B and C be nonzero integers. Let p, q and r be positive integers for which < 1. Then the generalized Fermat p q r equation Ax p + By q = Cz r has only finitely many solutions in integers x, y and z with gcd(x, y, z) = 1. 2

3 Lemma 2.3. Given integers P > 0 and r 3, there is a positive integer Y := Y (P, r) such that if k is an odd integer with k > Y and n is a positive integer, then k r 2 n 1 has a prime factor that is greater than P. Proof. Fix P and r as in the statement of the lemma. It suffices to show that there are only finitely many ordered pairs (k, n) with k odd such that k r 2 n 1 = p f 1 1 p ft t, (1) where the p i are all the distinct primes with p i P, and the f i are nonnegative integers. Suppose (1) holds with k odd. We put n = rn 1 + n 0 and f i = ru i + v i for each i {1, 2,..., t}, where n 1, u 1, u 2,... u t Z and n 0, v 1, v 2,..., v t {0, 1, 2,... r 1}. Then, we can rewrite (1) as p v 1 1 p vt t (p u 1 1 p ut t ) r 2 n 0 (2 n 1 k) r = 1. (2) Observe that, for r 3, the sum of the reciprocals of the generalized Fermat equation Ax r By r = Cz r+1, (3) where A, B and C are nonzero integers, is < 1. Thus, by Theorem 2.2, (3) has only finitely many solutions in integers x, y and z, with gcd(x, y, z) = 1. In particular, we let A = p v 1 1 p vt t, B = 2 n 0 and C = 1, and also suppose that z = 1. Then there are r t possibilities for A, depending on the v i, and r possibilities for B depending on n 0. We deduce that there are only finitely many possibilities for p u 1 1 p ut t and 2 n 1 k in (2) and, consequently, at most finitely many pairs (k, n) satisfying (1). Remark 2.4. An argument similar to the proof of Lemma 2.3 can be used to show that, given integers P > 0 and r 2, there is a positive integer Y := Y (P, r) such that if k is an odd integer with k > Y and n is a positive integer, then k r 2 n has a prime factor that is greater than P [5]. 3. The Proof of Theorem 1.1 To facilitate the building of a covering that can be used to establish Theorem 1.1, we prove and apply an extension of a group-theoretic result employed in [6]. Unlike the situation in [6], this extension has the advantage of being independent of the residues in our covering. 3

4 Lemma 3.1. Let m, r > 1 be integers such that gcd(m, r) = 1. Let p be an odd prime with p 1 (mod m), and let v (Z p ). If m 0 (mod ord p (v)), then there exists u (Z p ) such that v u r (mod p). In other words, v is an rth power modulo p. Proof. Let H = {x (Z p ) x m = 1}. Then H is a subgroup of (Z p ) with H = gcd(p 1, m) = m. Define a homomorphism θ : H H, by θ(h) = h r. Then the kernel of θ is precisely { h H h r = 1 } = {1}, since gcd(m, r) = 1. Hence, θ is an automorphism, and there exists u H such that v = θ(u) = u r. The following corollary, which is an immediate consequence of Lemma 3.1, is useful in our situation. Corollary 3.2. Let m > 1 be an integer with m 6, and let p be a primitive divisor of 2 m 1. If r is an integer with gcd(m, r) = 1, then 2 is an rth power modulo p The Proof of Theorem 1.1 Let C be a covering and let P be the set of all primes associated to C. What we deduce from Corollary 3.2 is that, given (a i, m i ) C with corresponding prime p i P, and a positive integer r with gcd(m i, r) = 1, there exists u i (Z pi ), such that u r i 2 a i (mod p i ). Thus, u r i 2 n 1 is divisible by p i whenever n a i (mod m i ). Therefore, Corollary 3.2 suggests that to maximize the density of the r-values captured, we should build C in which the number of distinct prime divisors of L C is minimized. This is precisely our strategy. We build our covering C with L C = = To determine how many times a particular modulus can be used in our covering, we first refer to the tables at [7, 8, 10]. These tables contain primitive divisors for numbers of the form 2 m 1, for certain values of m. If no information is given in these tables for a particular modulus m that we choose to use in our covering, then we simply use that particular modulus exactly one time since we are guaranteed to have at least one primitive divisor by Bang s theorem. Some of the prime divisors for 2 m 1 are not explicitly listed in the tables at [7, 8, 10], and since the primes that are actually listed are sometimes quite large, we refrain from providing them here. However, we do list below the congruences in our covering C, which was calculated using Magma and Maple. We point out that not every modulus m is used the maximum number of allowable times. For example, has nine 4

5 primitive divisors, but we only use the modulus m = 1225 eight times since using it more times is redundant with the choice of residues in C. Indeed, the covering C is minimal in the sense that no proper subset of C is a covering. The covering C contains 93 congruences and is C ={(1, 3), (4, 5), (0, 7), (2, 9), (3, 15), (9, 21), (0, 25), (20, 25), (26, 27), (12, 35), (31, 35), (23, 45), (32, 45), (20, 49), (39, 63), (60, 63), (15, 75), (30, 75), (44, 81), (71, 81), (17, 81), (36, 105), (57, 105), (87, 105), (8, 135), (116, 135), (131, 135), (111, 147), (110, 175), (85, 175), (10, 175), (18, 189), (81, 189), (140, 225), (5, 225), (176, 225), (131, 225), (116, 243), (35, 243), (197, 243), (8, 243), (6, 245), (181, 245), (201, 315), (96, 315), (306, 315), (62, 405), (332, 405), (305, 405), (321, 441), (342, 441), (174, 441), (195, 441), (447, 525), (132, 525), (342, 525), (510, 525), (446, 675), (311, 675), (221, 675), (86, 675), (531, 735), (132, 735), (447, 735), (321, 735), (636, 735), (237, 735), (522, 945), (711, 945), (575, 1215), (170, 1215), (491, 1215), (410, 1225), (1110, 1225), (760, 1225), (235, 1225), (60, 1225), (585, 1225), (935, 1225), (1077, 1225), (860, 1575), (1535, 1575), (491, 1575), (1391, 1575), (1166, 1575), (716, 1575), (167, 1701), (27, 2205), (2246, 2835), (41, 4725), (1812, 11025), (3191, 14175), (26816, 42525)}. Next, for any fixed r with gcd(r, 105) = 1, we apply the Chinese remainder theorem to the set of congruences { } {k 1 (mod 2)} k u i (mod p i ) i = 1,..., 93 to get infinitely many odd positive integers k in arithmetic progression, such that, for each integer n 1, we have that k r 2 n 1 is divisible by some prime in P. Since k r 2 n 1 > p i P for all i and n, the first part of the theorem is established. By Lemma 2.3, with P equal to the largest element in P, we deduce, when r 4, that there exist infinitely many values of k in our arithmetic progression such that k r 2 n 1 has at least two distinct prime divisors, and the proof of the theorem is complete. 5

6 4. Comments and Conclusions One of the first questions that comes to mind is: Can we do better?. In that regard, the techniques used in this paper contain an inherent weakness. Obviously, we cannot create a covering C such that L C has no prime divisors. Thus, we will always have holes in the set of values of r we are able to achieve using a single covering. We could attempt to patch these holes using multiple coverings with different values of L C, but there will still be holes corresponding to the overlap of the missing values in each situation. For example, using the covering C given in the proof of Theorem 1.1, we have captured r = 22 but not r = 6. If we can find a second covering C that captures all values of r with gcd(r, 11) = 1, but not values of r 0 (mod 11), then from C alone we have captured r = 6 but not r = 22. Together, of course, we have captured both r = 6 and r = 22, but we still have not captured r = 66. Aside from this drawback, it seems unlikely that we can achieve a better result using a single covering C C. Here, better means that either L C is odd and has exactly two distinct prime divisors, or L C is odd and has exactly three distinct prime divisors other than 3, 5 and 7. One difficulty in constructing C arises from the lack of the availability of enough primitive divisors, since repeated moduli must be used to compensate for the restrictions on the divisors of the moduli. Although we cannot provide a proof, we conjecture, for example, that it is impossible to construct a covering C satisfying both of the following conditions: the only prime divisors of L C are 3 and 5 each modulus m in C can be used as many times as there are primitive divisors of 2 m 1. References [1] A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift for Mat. 5(4) (1886), 70 80, [2] Y.-G. Chen, On integers of the forms k r 2 n and k r 2 n + 1, J. Number Theory 98 (2003), [3] H. Darmon and A. Granville, On the equations z m = F (x, y) and Ax p + By q = Cz r, Bull. London Math. Soc. 27 (1995),

7 [4] P. Erdős, On integers of the form 2k + p and some related problems, Summa Brasil Math, 29 (1949), [5] M. Filaseta, C. E. Finch, and M. R. Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac s conjecture, J. Number Theory 128 (2008), no. 7, [6] C. Finch, J. Harrington and L. Jones, Nonlinear Sierpiński and Riesel numbers, J. Number Theory 133 (2013), no. 2, [7] [8] P. Leyland, cunningham/2-.txt. [9] H. Riesel, Några stora primtal, Elementa 39 (1956) [10] S. S. Wagstaff, pmain814. 7