From: gerry@mpce.mq.edu.au (Gerry Myerson) Subject: Neat stuff in Math. Comp. Date: Thu, 25 Feb 1999 16:43:51 +1100 Newsgroups: sci.math Keywords: Mixed bag... The January, 1999 issue of Mathematics of Computation (Vol. 68, # 225) has many articles on topics that come up on this news group from time to time, to wit: 1. Pierre L'Ecuyer, Tables of linear congruential generators of different sizes and good lattice structure, pp 249-260, and Tables of maximally equidistributed combined LFSR generators, pp 261-269. These are about (pseudo-)random number generators. 2. Tomas Oliveira e Silva, Maximum excursion and stopping time record-holders for the 3x + 1 problem, 371-384. Everyone who has played around with the 3x + 1 problem knows that if you start with 27 it takes a long time to get to 1 & you reach some scary big numbers before you get there. 27 is a record-holder for both measures, in that no smaller starting point takes as long to reduce, and no smaller starting point goes up so high. This paper has all the record-holders up to 3 x 2^53, which is about 2.7 x 10^16. To my surprise, once you get past 27, the lists of the two kinds of record- holders are mostly disjoint; only 703 and 270271 appear on both lists. You might try these starting points: 319 804 831; 3 716 509 998 199; 12 235 060 455; 1 008 932 249 296 231. The paper also verifies the standard conjecture out to 3 x 2^53, and the author remarks that "the computer...is still running." 3. Manuel Benito, Juan L Varona, Advances in aliquot sequences, 389-393. Let s(n) be the sum of the proper divisors of n. What happens when you calculate s(n), s(s(n)), s(s(s(n))), etc.? Do there exist n for which this sequence goes to infinity? The smallest n for which the answer is not known is n = 276; after 913 iterations, the author reached a 90-digit number, and gave up. If you start with 3556, you reach 1 after 2058 iterations, with intermediate results as big as 75 digits; start with 4170, and you have to deal with an 84-digit number before you come down to 1. 564 was abandoned after 2230 iterations gave a 91-digit number, and 2514 was abandoned after 2794 iterations gave an 80-digit number. A couple of websites are given: www.unirioja.es/dptos/dmc/jvarona/aliquot.html www.loria.fr/~zimmerma/records/aliquot.html 4. Todd Cochrane, Robert E Dressler, Gaps between integers with the same prime factors, 395-401. Conjecture (Dressler). Between any two positive integers having the same prime factors there is a prime (verified up to larger of the two numbers less than 7 x 10^13). Conjecture. For every positive e there's a constant C(e) such that if a < c have the same prime factors then c - a > C(e) a^(.5 - e). Theorem. The abc conjecture implies the 2nd conjecture. They give relations to theorems and conjectures on gaps between consecutive primes. They also ask: Do there exist infinitely many solutions of 0 < p^a q^b - p^c q^d < p^(c/2) q^(d/2), p, q prime, a, b, c, d positive integers? Is there any a < c with the same prime factors such that c - a < a^(1/3)? 5. Miodrag Zivkovic, The number of primes sum from i = 1 to n of (-1)^(n - i) i! is finite, pp 403-409. If A_(n + 1) = n! - (n - 1)! + (n - 2)! - ... +/- 1!, and p = 3612703, then A_p is a multiple of p, so A_n is a multiple of p for all n > p, so A_n isn't prime for any n > p. Also, if !n = (n - 1)! + ... + 1!, then !26541 is not squarefree, as it's divisible by 54503^2. 6. Pierre Dusart, The k-th prime is greater than k(log k + log log k - 1) for k >= 2, pp 411-415. The title says it all. 7. Harvey Dubner, Wilfrid Keller, New Fibonacci and Lucas primes, pp 417-427. Prime and probable-prime numbers among the first 50000 terms of the Fibonacci and Lucas sequences. 8. Richard P Brent, Factorization of the tenth Fermat number, pp 429-451. A good introduction to state-of-the-art factorization & primality-testing techniques, with an extensive bibliography. Gerry Myerson (gerry@mpce.mq.edu.au)