From Herman.te.Riele@cwi.nl Fri Feb 14 20:40:12 1997
Date: Fri, 14 Feb 1997 13:04:28 -0500
From: Herman.te.Riele@cwi.nl
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: New amicable pairs of record size

ANNOUNCEMENT

Two large amicable pairs of record size (1739 resp. 1923 decimal digits)
were found by Mariano Garcia (New York) and sent to Herman te Riele on
Febr. 10, 1997.
Mariano Garcia hopes to celebrate his 80-th birthday in 1998.
No less than fourty years ago he already discovered new amicable pairs
(published in Scripta Mathematica, vol. 23, 1957, pp. 167--171).

(H. te Riele is working on proving primality of the large strong pseudo-primes
q1 and q2 below with a Lucas-Lehmer test, q_i + 1 being easy to factorize).

First amicable pair, of type (7,2):
-----------------------------------
a1 = 3 * 5 * 7 * p^n * m * q1 and
a2 = 3 * 5 * 7 * p^n * q * q2
p = 5782469641007794140718386001 (prime) n = 30
m = 2620181556081656432875296001
  = 11 * 13 * 37 * 3779 * 19994749 * 6553914555541
q = 28856427635109598805175661612659605699685119 (prime)
q1 = h1 * p^n - 1 where h1 = p + q
(# decimal digits of q1 is 877;
 q1 is a strong pseudo-prime for ten randomly chosen bases)
q2 = h2 * p^n - 1 where h2 = p - m
(# decimal digits of q2 is 861;
 q2 is a strong pseudo-prime for ten randomly chosen bases)
# decimal digits of both a1 and a2 is 1739

Second amicable pair, of type (7,2):
------------------------------------
a1 = 3 * 5 * 7 * p^n * m * q1,
a2 = 3 * 5 * 7 * p^n * q * q2,
p = 1615214022515684912115550808251201 (prime) n = 28
m = 2620181556081656432875296001
  = 11 * 13 * 37 * 3779 * 19994749 * 6553914555541
q = 3162299405964746117411039999 (prime)
q1 = h1 * p^n - 1 where h1 = p + q
(# decimal digits of q1 is 964;
 q1 is a strong pseudo-prime for ten randomly chosen bases)
q2 = h2 * p^n - 1 where h2 = p - m
(# decimal digits of q2 is 964;
 q2 is a strong pseudo-prime for ten randomly chosen bases)
# decimal digits of both a1 and a2 is 1923

Previous record size amicable pairs
(the largest is listed below, having 1478 decimal digits)
were found by Frank Zweers from the University of Dortmund, Germany,
and sent to Herman te Riele on Dec. 4, 1997.

Amicable pair of type (5,2):
----------------------------
a1 = 2^10 * p^n * m * q1,
a2 = 2^10 * p^n * q * q2,
p = 119927097385576139002079233 (prime) n = 27
m = 59934255312869542712448001
  = 1087 * 17509 * 2580653 * 1220266291199
q = 16870978894719348647574068358773735423999 (prime)
q1 = h1 * p^n - 1 where h1 = p + q
q2 = h2 * p^n - 1 where h2 = p - m
# decimal digits of both a1 and a2 is 1478