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On a Sequence Arising in Series for It mathematics of computation volume 42, number 165 january 1984, pages 199 217 On a Sequence Arising in Series for it By Morris Newman* and Daniel Shanks Abstract. In a recent investigation of dihedral quartic fields [6] a rational sequence {a„} was encountered. We show that these a„ are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding ^-adic numbers are known completely [2]. Other unsolved problems are presented. The growth of the a„ is examined and a new algorithm for computing a„ is given. An appendix by D. Zagier, which carries the investigation further, is added. 1. Introduction. The sequence (an) that begins with (1) a{ = 1, a2 = 47, a3 = 2488, a4 = 138799, a5 = 7976456, a6 = 467232200, and which is defined below, is encountered in a set of remarkable convergent series for it. These are (see [6]): (2) W = ^-(-.og|i/|-24£(-l)^c/«), where A7is a positive integer and U = U(N) is a real algebraic number determined by N. Some of these series are remarkable because of their almost unbelievably rapid rates of convergence. For example, for N = 3502, (2) converges at 79 decimals per term and its leading term, namely --¿¿=rlOg£/, /35ÖT differs from m by less than 7.37 • 10~82. In this case, (3) t/ = U(35Q2)= (2defg)~b, where (4) d= D + ]/D2 - 1 , e = E + JE2 - 1 , f=F+]/F2-\, g = G+Jg2 - 1 , for the quadratic surds D = |(1071 + 184v/34), E = ¿(1553 + 266v/34), F = 429 + 304v/2, G - ^(627 + 442/2 ). In this example, the six an in (1) already give m correctly to over 500 decimals. Received September 9, 1982. 1980Mathematics Subject Classification. Primary 10D05,10A35, 10B40. *The work of the first author was supported by the National Science Foundation. © 1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 199 200 MORRIS NEWMAN AND DANIEL SHANKS For N = 2737, and the more general (6) U=(-\)N(2defg)\ the quadratic surds 0 = 4(621+49/161), E = ¿(321 + 25/Ï6Î), (7) F = ¿(393 + 31/Í6T), G = ¿(2529+ 199/Ï6Ï), and (4) unchanged, define its negative value of (7(2737). Now (2) converges at only 69 decimals per term. See [6] for other examples of even and odd N, and the corresponding positive and negative values of U, where (2) also converges very rapidly. The definition given in [6] of an is rather complicated. We have a relation 00 (8) i/= V\~\(\ + Vf* n-1 between our U = U(N) and the number (9) V=V(N) = (-l)Ne~^N. The inversion of (8) gives F as a power series in U: (io) k= t(-\y-]cnu" n- 1 that begins with c, = 1, c2 = 24, c-}= 852,_Now, in the power series for (ii) log(A0 + Kn)}= K+ T + '"' substitute (10), and thereby define an recursively by (i2) E(-i),"l-f£/- = iog no + v»)\. n.i " U-i j Then, the logarithm of (8) gives us (2). In [6], only the six coefficients in (1) were given, since they were computed by hand, a tedious operation. (The original an so computed contained an error which was discovered when R. Brent kindly attempted to verify (2) for N = 3502 to the aforementioned 500 decimals.) Clearly, the an are best calculated using a digital computer. The first 100 values of an and cn were so computed in about 8 minutes. The first 50 values of a„ and c„ are given in Tables 1 and 2. 2. Properties of an. A. We observe that all a„ in Table 1 are positive integers. It was obvious from the recursion above that the an are rational but not that they are positive and integral. However, we prove below that 00 (13) 24an is the coefficient of x" in \~\ (1 + x2*"1)24", k=\ which implies that an is a positive integer. B. We observe that all an in Table 1 satisfy (14) an is odd if and only if n is a power of 2. ON A SEQUENCEARISING IN SERIES FOR ■n 201 This unexpected result is reminiscent of C. R. Johnson's conjecture for the parity of the number of subgroups of the classical modular group of a given index N, see [7]. That conjecture was proved by Stothers and, independently, by A. O. L. Atkin. The present observation (14) is proved below. C. A striking paradox about this proven (14) for the parity of an is this: As presented above, the cn in (10) would appear to constitute a simpler sequence than our an in (12), since its definition is much more direct. Nonetheless, we have been unable to determine the parity of c„. In Table 2 one readily observes that (14a) c„ is odd only when « = 8/c + 1 and is odd if k = 0, 1, 2, 4, 6. But what are these kl We do not know, and do not even have a conjecture for the parity of c„. It is easy to prove (14a) and to compute c„ modulo 2. The parity of cn appears to be random with increasing k just as is the parity of the unrestricted partition functionp(n). (See [8] for the latter.) As for the claim above that we have a paradox here, see Zagier's comment in the appendix. D. A second, more important paradox concerns an modulo 3. We conjectured (15) a„5É0mod3 for all n. While (15) appears simpler than (14), we did not prove it. Every positive integer n has a unique representation (16) » = 3*(3m±l) with nonnegative k, m. A stronger conjecture than (15) is (17) Û3*(3m±i)= ±lmod3. For greater clarity, let us rewrite (17) as follows: (18a) a3m+1 = lmod3, (18b) a3m_x = -1 mod3, (18c) a3m = ammod3. These are clearly equivalent to (17). We did not prove the simple-looking (18a) and (18b). The more subtle-looking (18c) we did prove; it is a simple corollary of a much more general congruence given in E below. We did verify ( 17) up to a,43 = -1 mod 3 by computer, and we both believed it to be true. After we finished the first version of this paper, we showed the conjecture to D. Zagier, and, as we expected, he proved it. See the appendix. E. The important general congruence alluded to above, and proved below, is (19) VEV-,mod^' valid for every prime p and all positive integers m and k. For k = 1 this gives us (20) amp = am mod p and (18c) is obviously the case/? = 3. Congruence (20) is computationally useful. For example, what is 094 modulo 94? Since a2 47 = a2 = 47 mod 47, 202 MORRIS NEWMAN AND DANIEL SHANKS we have a94 = 0 mod 47. But also a94 = 0 mod 2, by ( 14). Therefore a94 = 0 mod 94. Similarly, we can evaluate a2p modulo 2p for any prime p, and in particular we see that, for any prime p, (21) a2p * 1 mod2/7. F. The choice m = 1 in (20) gives us (22) ap = a, = 1 mod p, which we call the Fermât Property. It is a necessary condition for primality. Of course, we ask: Is (23) an = 1 mod n, n > 1, a sufficient condition for primality? We have just seen in (21) that n = 2p can never satisfy (23). But consider a3 = 2488 = 3 ■829 + 1. Since 829 is prime, we have by (20) that a24S7 - <*3- 1 mod 829. and similarly a2487 = a829 mod 3. But 829 = 3m + 1, and since (18a) is now true, we also have (24) a24(t7= 1 mod 3. Then (23) holds for the composite 2487 = 3 • 829. So (23) is not a sufficient condition for primality. Even if it were, it would not be a practical test for primality. The calculation of an modulo n requires at least O(n) operations by any algorithm known to us. G. We return to (19) and specialize in a different direction; m = 1 gives us (25) apí = apí i mod pk. Fix p and consider the sequence (26) (a,* modulo/?*}, k = 1,2,3. If we write these numbers to the base/?, (25) guarantees that each time k is increased by 1, and we add one more /?-adic digit on the left, all the earlier p-adic digits on the right remain unchanged. Thus, for each p, the sequence (26) defines a /?-adic number. For example, for p = 2, (26) begins (in decimal) as 1, 3, 7, 15, 15, 47. and so we have the 2-adic number (reading from right to left) ...1000101111. (base 2). Similarly, for/? = 3 and 5, we have ...0111. (base 3) ... 4 11. (base 5). But what are these p-adic numbers? We do not know. Are they algebraic or transcendental? We do not know. Contrast this ignorance with the situation in I below. ON A SEQUENCE ARISING IN SERIES FOR ir 203 We do have, for every /?, (27) api = 1 + p mod p2, so the first two /?-adic digits on the right are both 1. The first 1 follows from the Fermât Property (22) but the second 1 does not follow from the general congruence ( 19), and again contrasts with the situation in I below. This (27) was first proved by our colleague L. Washington. Our proof below is different. Perhaps we should note that the sequence (28) {V}, k= 1,2,3,... defines the same /?-adic number that (26) does. The latter looks a little simpler since it adds exactly one/?-adic digit each time.
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