The hunt for the largest Carmichael numbers has begun ... some 20+ years ago

[Neptune]

... 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):
3-C11219

It's construction-method:

{
x=vecprod(primes(80))/2;    /* product of first 79 odd primes */
/* choose 'i' such that 'p' and 'q' are primes */
i = 74218118677881697744516449513018846424418844889229;
m = (x*i-1)^13/4;
p = 6*m+1;
q = 12*m+1; 
r = (p*q-1)/(5*7*11*13)+1;
c = p*q*r;          /* todo: ECPP-certificates 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(C-vector(#C,x,1))==1};

isc([p,q,r])
1

#digits(c)
11219

Still need to prove the primality of p,q and r. Any one can help with providing an ECPP-certificate?

[R. Gerbicz]

Originally Posted by Neptune

... right here - right now!

The largest Carmichael number with 3 factors currently is (unless otherwise stated):
3-C11219

That would be too trivial, maybe that could be the record in another century.

Originally Posted by Neptune

Still need to prove the primality of p,q and r. Any one can help with providing an ECPP-certificate?

I would suggest to search in a form, where p-1,q-1,r-1 number's factorization is known.

[Batalov]

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 32-bit 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 32-bit OS images.

[R. Gerbicz]

Originally Posted by Neptune

The largest Carmichael number with 3 factors currently is (unless otherwise stated):
3-C11219

Well, I know it is a trivial reference, larger records are:
https://mathworld.wolfram.com/CarmichaelNumber.html
And found not any better. If there is one, then they not mentioned this 60351 digits number.

[Batalov]

NMBRTHRY list is searchable

Some hits over different years:
3-Carmichael number with 61072 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R458
4-Carmichael number with 30366 digits https://listserv.nodak.edu/cgi-bin/w...=NMBRTHRY&P=R2
5-Carmichael number with 14241 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R747
6-Carmichael number with 20961 digits https://listserv.nodak.edu/cgi-bin/w...NMBRTHRY&P=R29
7-Carmichael number with 3773 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R564
8-Carmichael number with 16432 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R574
9-Carmichael number with 11310 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R564
19-Carmichael number with 23707 digits https://listserv.nodak.edu/cgi-bin/w...MBRTHRY&P=R527

[Neptune]

Originally Posted by Batalov

NMBRTHRY list is searchable

Some hits over different years...

Thank you for the list!

I see, my modest 11k-digit 3-Carmichael number comes a bit late as D. Broadhurst already had a 60k-digit in 2002.

Better for me at the moment to hold on with the Erdős method:
99-Carmichael number with 23415 digits 99-C23415.

[paulunderwood]

Originally Posted by Neptune

Better for me at the moment to hold on with the Erdős method:
99-Carmichael number with 23415 digits 99-C23415.

Factors factored out and reported to FactorDB

[Neptune]

Originally Posted by Neptune

Still need to prove the primality of p,q and r. Any one can help with providing an ECPP-certificate?

An unknown user worked out the ECPP-certificates for p and r between May 22/23 and uploaded them to factordb.com. Many thanks for your efforts!
Together with the successful N+1 test for q, which I missed completely, 3-C11219 is now a proven Carmichael number.

Originally Posted by R. Gerbicz

I would suggest to search in a form, where p-1,q-1,r-1 number's factorization is known.

I agree with this. For larger Carmichael numbers, we cannot afford to spend more time on the ECPP of the factors than constructing the former.
With my current method the factors of p-1 can be handled quite well. My new record:

81-Carmichael number with 23883 digits: 81-C23883

[paulunderwood]

Originally Posted by Neptune

An unknown user worked out the ECPP-certificates for p and r between May 22/23 and uploaded them to factordb.com. Many thanks for your efforts!
Together with the successful N+1 test for q, which I missed completely, 3-C11219 is now a proven Carmichael number.

'twas I

I agree with this. For larger Carmichael numbers, we cannot afford to spend more time on the ECPP of the factors than constructing the former.

Surely a job Serge+Ryan to tackle.

With my current method the factors of p-1 can be handled quite well. My new record:

81-Carmichael number with 23883 digits: 81-C23883

Factored for FactorDB