From 70372.1170@CompuServe.COM Mon Jun 9 15:06:28 1997
Date: Mon, 9 Jun 1997 07:56:39 -0400
From: Harvey Dubner <70372.1170@CompuServe.COM>
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: Record Amicable numbers
Mariano Garcia has found four very large Amicable pairs (AP) in recent
weeks. The size of these new pairs are 2332 digits, 3193 digits, 3187
digits and 3383 digits. We believe that the last three are the largest
known Amicable numbers. What is truly impressive is the current rapid
increase in the size of the record .
Prior to 1974 the largest known AP was 25 digits. In 1974 Herman te Riele
found an AP of 152 digits, and later in 1983 an AP of 282 digits. In 1988
Roger Wiethaus discovered a 1041 digit AP. In December, 1996 Frank Zweers
found an AP of 1478 digits. On February 10, 1997, Mariano Garcia found AP's
with 1739 and 1923 digits. In April, 1997 Zweers found AP's with 1827,
2030 and 2725 digits. Now Garcia has raised the record to 3383 digits.
The method used by Garcia is based on the Master's thesis of Roger
Wiethaus and Garcias paper published in the Journal of Recreational
Mathematics, 1993, "Favorable Conditions for Amicability." Here are his
four new AP's.
1. Found April 23, 1997 - 2332 digits
p = 1570664039150220631 prime
q = 11480325216000351479 prime
m = 37*197*58313*1700173831
q1 = (p+q)*p^63 - 1 prime
q2 = (p-m)*p^63 - 1 prime
The Amicable pair:
A1 = 2^5 * p^63 * m * q1
A2 = 2^5 * p^63 * q * q2
----------------------------------------
2. Found May 4, 1997 - 3193 digits
p = 260249770826245373300051 prime
q = 59571442684898401985125884107 prime
m = 569*5039*1479911*30636732851
q1 = (p+q)*p^67 - 1 prime
q2 = (p-m)*p^67 - 1 prime
The Amicable pair:
A1 = 2^9 * p^67 * m * q1
A2 = 2^9 * p^67 * q * q2
---------------------------------------
3. Found May 12,1997 - 3187 digits
p = 37668608243682748398344917 prime
q = 95269476522053875552478915078194367039 prime
m = 569*5023*22866511*287905188653
q1 = (p+q)*p^61 - 1 prime
q2 = (p-m)*p^61 - 1 prime
The Amicable pair:
A1 = 2^9 * p^61 * m * q1
A2 = 2^9 * p^61 * q * q2
---------------------------------------
3. Found May 20,1997 - 3383 digits
p = 37669773212168992472511541 prime
q = 609610904872320606430695102719 prime
m = 569*5023*22866511*287905188653
q1 = (p+q)*p^65 - 1 prime
q2 = (p-m)*p^65 - 1 prime
The Amicable pair:
A1 = 2^9 * p^65 * m * q1
A2 = 2^9 * p^65 * q * q2
The above searches were done on a 486/33 equipped with a Dubner Cruncher.
Finding the four AP's took about a Cruncher/month of computer time.
Mariano Garcia does not have email. I will be happy to forward any
messages or questions. Also, if anyone wants information on the Cruncher
just ask.
Harvey Dubner