# 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.
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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