3-component Carmichael Numbers



Numbers of the form,

(6M+1)(12M+1)(18m+1)

are Carmichael numbers whenever all three components are prime.  This,
perhaps, is the best known family of Carmichael numbers.  Because of
the single parameter that appears in each of the components, the form
of M can be chosen so that prime proving is fast and easy.  Also,
there are more Carmichael numbers in this family than any other.
There is data indicating that about 2.5% of all 3-component Carmichael
numbers are of this form.

The smallest such Carnichael number is 7*13*19=1729.  I have recently
found the three largest such numbers (as far as I know).  They are of
the form,

N=P_1*P_2*P_3  where   P_i=c*3003*k_i*10^b + 1,   k_i=1,2,3

             Times based on pentium/200 equivalent

b=exp  c=multiplier  digits  Estimated search time  actual time
__________________________________________________
1502    6948950     4538  131 computer-days    12 days  (lucky!!)
1308    19513527   3958    33 computer-days    36 days
1204    51412393   3647    23 computer-days    48 days


Note that each component of the number with 4538 digits has 1513 digits.
Finding 3 primes simultaneously of this size is not easy.

I also counted the number of such Carmichael numbers of this type up to
10^36.

----------------------------------------------------------------------------
3 component Carmichael numbers of the form
      C = (6M+1)(12M+1)(18M+1)

  n     Number of C's < 10^n
-----   ----------------------
3          0
4          1
5          1
6          2
7          2
8          3
9          7
10         10
11         16
12         27
13         45
14         77
15        133
16        234
17        415
18        746
19       1354
20       2480
21       4580
22       8519
23      15956
24      30069
25      56988
26     108570
27     207836
28     399638
29     771621
30    1495580
31    2909178
32    5677865
33   11116339
34   21828157
35   42670184
36   27940603

[Added later: See the following correction.]

I know that Wilfrid Keller previously had counted up to 10^30 some years
ago.  The new, fast computers are wonderful.

Harvey Dubner