20230520, 18:43  #1 
"Martin Hopf"
Jul 2022
Solar System
101 Posts 
The hunt for the largest Carmichael numbers has begun ... some 20+ years ago
... right here  right now!
These numbers were introduced by the namesake as "absolute pseudoprimes". Isn't that a reason, to pay more attention on them? Don't be blinded by big Carmichael numbers with many factors, as these are always easy to construct. Concentrate on Carmichael numbers with few factors. Let's say less than a dozen or a hundred. The largest Carmichael number with 3 factors currently is (unless otherwise stated): 3C11219 It's constructionmethod: Code:
{ x=vecprod(primes(80))/2; /* product of first 79 odd primes */ /* choose 'i' such that 'p' and 'q' are primes */ i = 74218118677881697744516449513018846424418844889229; m = (x*i1)^13/4; p = 6*m+1; q = 12*m+1; r = (p*q1)/(5*7*11*13)+1; c = p*q*r; /* todo: ECPPcertificates for 'p,q' and 'r' */ } /* diagnostic follows */ ispseudoprime([p,q,r]) [1, 1, 1] /* from now on 'p,q,r' are titled as strong probable primes! */ /* isc(C) returns '1' if the product of all primes in vector 'C' is a Carmichael number. */ isc(C)={vecprod(C)%lcm(Cvector(#C,x,1))==1}; isc([p,q,r]) 1 #digits(c) 11219 Last fiddled with by Neptune on 20230520 at 19:04 
20230520, 19:33  #2  
"Robert Gerbicz"
Oct 2005
Hungary
23·71 Posts 
Quote:
I would suggest to search in a form, where p1,q1,r1 number's factorization is known. 

20230520, 19:53  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2×13×17×23 Posts 
A version of (R. Gerbicz') polysieve will find much larger numbers in a very short time.
Or one can make a prototype (as some people say) "on their knee". For example: Step 1: sieve for CC len 2 in base 2; these are conveniently already a pair of 6k+1 and 12k+1 right away (where k=2^{m}). Step 2: either sieve the result with triple leading coefficient  or run pfgw f on triple leading coefficient and then recover the underlying CC pairs, and done. Or one can modify NewPGen  but it only compiles on ancient 32bit configured machines and with GCC version <=3. I used to do that for the quad forms that I liked, but now I am not sure even AWS has 32bit OS images. 
20230521, 02:01  #4  
"Robert Gerbicz"
Oct 2005
Hungary
23·71 Posts 
Quote:
https://mathworld.wolfram.com/CarmichaelNumber.html And found not any better. If there is one, then they not mentioned this 60351 digits number. 

20230521, 07:11  #5 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23666_{8} Posts 
NMBRTHRY list is searchable
Some hits over different years: 3Carmichael number with 61072 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R458 4Carmichael number with 30366 digits https://listserv.nodak.edu/cgibin/w...=NMBRTHRY&P=R2 5Carmichael number with 14241 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R747 6Carmichael number with 20961 digits https://listserv.nodak.edu/cgibin/w...NMBRTHRY&P=R29 7Carmichael number with 3773 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R564 8Carmichael number with 16432 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R574 9Carmichael number with 11310 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R564 19Carmichael number with 23707 digits https://listserv.nodak.edu/cgibin/w...MBRTHRY&P=R527 
20230521, 20:07  #6 
"Martin Hopf"
Jul 2022
Solar System
101 Posts 
Thank you for the list!
I see, my modest 11kdigit 3Carmichael number comes a bit late as D. Broadhurst already had a 60kdigit in 2002. Better for me at the moment to hold on with the Erdős method: 99Carmichael number with 23415 digits 99C23415. 
20230525, 18:00  #8  
"Martin Hopf"
Jul 2022
Solar System
101 Posts 
Quote:
Together with the successful N+1 test for q, which I missed completely, 3C11219 is now a proven Carmichael number. Quote:
With my current method the factors of p1 can be handled quite well. My new record: 81Carmichael number with 23883 digits: 81C23883 Last fiddled with by Neptune on 20230525 at 18:27 

20230525, 20:36  #9  
Sep 2002
Database er0rr
11072_{8} Posts 
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