From 70372.1170@CompuServe.COMWed May 1 17:38:25 1996
Date: Wed, 1 May 1996 09:30:57 EDT
From: Harvey Dubner <70372.1170@CompuServe.COM>
To: Multiple recipients of list NMBRTHRY
Subject: Carmichael number record
In the past several years there has been a welcome resurgence of interest
in Carmichael numbers. Some wonderful theory has been developed including
the proof that there are an infinite number of Carmichael numbers. An
excellent summary of the past history and current status of this subject
appears in Paul Ribenboim's book, "The New Book of Prime Number Records"
published by Springer, 1996.
In 1939 Chernick derived one-parameter expressions for Carmichael numbers
which he called "Universal Forms," the most prominent of these being
U3 = (6M + 1)(12M + 1)(18M + 1).
U3 is a Carmichael number when the quantities in parentheses are
simultaneously prime. Until 1988 the largest known Carmichael numbers
were of this form. Since then new theory has resulted in Carmichael
numbers with millions of digits and a 3-component Carmichael number of a
special form with 8030-digits.
However investigating U3 is still interesting. Wilfrid Keller has counted
all U3's up to 10^30. Until recently the largest known U3 had 1265 digits
which I found in the late 1980's. Hardware improvements and availability
(Dubner Crunchers) and better search methods made it reasonable to search
for a 3000+ digit U3 Carmichael number consisting of 3 components, each of
which is a Titanic prime.
Using one Cruncher the expected search time was about 30 days. The actual
search time was almost 50 Cruncher-days, using 3 Crunchers. After we
found one solution we accidently left a Cruncher running and a few days
later a second solution popped up. Thus the actual average search time
came close to the theoretial time. Nothing like a little luck to make
science come out right.
Here are the Carmichael numbers: U3 = P*Q*R
P = 23830007*3003*1*10^1013 + 1 , 1024 digits
Q = 23830007*3003*2*10^1013 + 1 , 1025 digits
R = 23830007*3003*3*10^1013 + 1 , 1025 digits
U3 = 3073 digits
P = 50762023*3003*1*10^1013 + 1 , 1025 digits
Q = 50762023*3003*2*10^1013 + 1 , 1025 digits
R = 50762023*3003*3*10^1013 + 1 , 1025 digits
U3 = 3074 digits
If anyone wants sales or technical information about the Cruncher, just
ask.
Harvey Dubner