The Largest Known CPAP-3
November 2004 a new record for the largest known CPAP-3 (3 consecutive primes in
arithmetic progression) was set by Torbjörn Alm,
Hans Rosenthal, Jens Kruse Andersen and Tom Kuechler, using the programs
PrimeForm/GW, Yves Gallot's Proth.exe, and Marcel Martin's Primo:
87 · 2^24582 - 1 + 1290n, for n = 0,1,2 (7402 digits)
When found, this was the largest known case of any 3 consecutive proven
primes, and there was no larger known CPAP-3 with probable primes.
(January 2005 David Broadhurst announced
two 7535-digit CPAP-3, the larger proved
in February by François Morain.)
The current CPAP records are at The Largest Known CPAP's.
The titanic records are also at the top twenty Consecutive Primes in Arithmetic Progression in Chris Caldwell's
The Prime Pages.
An announcement of this record and the used algorithm is on the Number Theory List.
· 2^24582 - 1 has a form which means it is easily proved prime, e.g. with PrimeForm/GW or
The certificate files are offline after a website move on January 23 2009 so
some of the below links are not working.
Torbjörn Alm proved 87
· 2^24582 + 1289 with Primo 2.2.0 beta 1 on an AMD Athlon XP 2500+ cpu
running at 1.9 GHz. The Primo certificate
http://hjem.get2net.dk/carlkruse/certif/cpap7402a.zip (3.2 MB)
The total running time was around 5000 hours.
Hans Rosenthal originally proved 87
· 2^24582 + 2579 with different Primo versions and
pc's. The certificate was valid but some problems meant there was no valid running time.
He later proved it again with Primo 2.2.0 beta 4 on an AMD Athlon XP 2400+ cpu running
at 2.0 GHz. This certificate is in http://hjem.get2net.dk/carlkruse/certif/cpap7402b.zip
The total running time was 3148 hours with more details in http://hjem.get2net.dk/carlkruse/certif/cpap7402b-time.txt
Both certificates have been verified by both Primo and Jim Fougeron's Cert_Val.
They are currently the 2nd and 3rd largest Primo
Based on experiments with an earlier Primo version, Mike Oakes once estimated the time for phase 1 is typically around proportional to bits^4.86.
There have been other estimates and Marcel Martin has never published an estimate to my knowledge. It is known that running time can vary a lot for candidates of the same size.
Torbjörn Alm kept a log of his Primo certification:
Primo implements the ECPP algorithm. Phase 1 repeatedly reduces the size of a number which must be proven prime to prove the original candidate.
The log has some lines like:
2004-03-18 18:31 Test 102 started 21745/24589 (~55% left)
21745/24589 means 21745 of the original 24589 bits are remaining at this time.
~55% left means around 55% of phase 1 is estimated to remain, because (21745/24589)^4.86 = 0.55.
Made by Jens Kruse Andersen,
Last updated 12 June 2014