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)