J. RECREATIONAL MATHEMATICS, Vol. 22(1), 2-6, 1990
CARMICHAEL NUMBERS WHICH
ARE THE PRODUCT OF THREE
CARMICHAEL NUMBERS
HARVEY DUBNER
Dubner Computer Systems, Inc.
6 Forest Avenue
Paramus, New Jersey 07652
HARRY NELSON
4259 Emory Way
Livermore, California 94550
Introduction
Fermat in 1640 discovered what is now known as his “Little Theorem,” which states:
If N is prime, and a is relatively prime to N, then aN-1 -1 is divisible by N.
In congruence notation, Fermat’s equation is:
(1)
a N 1  1 (mod N ).
N 1
The Chinese as early as 500 B.C. knew of this for a = 2. They also believed that if 2  1 is
divisible by N, then N is prime, and this converse of the theorem was generally accepted until
1819 when it was discovered that 2340 – 1 is divisible by 341 = 11 * 31, a composite number. A
few years later it was found that 91 = 7 * 13 divides 390 – 1. As the years went by, more and
more special cases were found where combinations of certain composite numbers, N, and bases,
a, satisified Fermat’s equation. Then, in 1907, Robert D. Carmichael proved there are composite
numbers which satisfy Fermat’s equation for all values of a which are relatively prime to N. The
three smallest Carmichael numbers are 561 = 3 * 11 * 17, 1105 = 5 * 13 * 17, 1729 = 13 * 19.
There are 1547 such numbers up to 1010.
Fermat’s Little Theorem is a thing of beauty: it is simple, important, and it is universal in the
sense that it states a property of all prime numbers – such general statements are difficult to find
and very few are known. It is a shame that the converse is not true, but the compensation for this
is the remarkable amount of theory that it spawned, much of it because of the existence of
Carmichael numbers.
© 1990,, Baywood Publishing Co., Inc.
CARMICHAEL NUMBERS
Although Carmichael numbers masquerade as prime numbers with respect to Fermat’s equation,
in other respects they behave quite differently. For instance, one Carmichael number may divide
another, as in the example:
CN1  7  13  19,
CN2  7  13  19  37.
Also, it has been known at least since 1948 [2] that a Carmichael number can be the product of
two other Carmichael numbers, as in
CN3  7  13  1937  73  109,
where each of the numbers in parentheses is a Carmichael number. The main purpose of this
article is to introduce Carmichael triples, which are the product of three other Carmichael
numbers.
Derivation
Chernick stated the following theorem for all Carmichael numbers [1]:
Fermat’s Little Theorem holds for composite integers if and only if N can be
expressed as a product of distinct odd primes P1 , P2 ...., Pn n  2  and each
Pi  1 divides N  1.
Chernak derived one parameter expressions for Carmichael numbers, the most prominent of
these being
(2)
N3  6M  112M  118M  1,
where M must be chosen such that the quantities in parentheses are prime. We will combine
three of these expressions whose product satisfied Chernick’s theorem and is also dependent on a
single parameter.
Let Pi  ai SM  1, i  1, 2, 3,..., r; Pi prime.
In the following equation, for notational ease, assume that
 ai a j means
r
r
 ai a j ,
i j
 ai a j ak means
r
a a a ,
i j k
i,
j
k
etc.
r
N   Pi   ai SM  1
 1  SM  ai  S 2 M 2  ai a j  S 3 M 3  ai a j a k  ...  S r M r  ai a j ak ...ar
Assume that S   ai
N  1  S 2 M 2  ai a j  S 3M 3 F


N  1  S 2 M 1  M  ai a j  SM 2 F  S 2 MG
N  1 S 2 MG SG


Pi  1 ai SM
ai
2
(3)
H. DUBNER & H. NELSON
If each ai divides SG, then N is a Carmichael number. Requiring that each a divides the sum S
leads to quite interesting results which is the subject of another paper in preparation. By
satisfying the somewhat less restrictive conditions of equation (3), it is possible to find
Carmichael triples. Finding appropriate values of a is rather like solving a puzzle. Equation (3)
must be satisfied simultaneously for each of the three components as well as for the product of
the three components.
For example:
N  N1 N 2 N 3 , and ai  1, 2, 3; 6, 12, 18; 14, 28, 42
N1  SM  1 2SM  1 3SM  1
N 2  6SM  1 12SM  1 18SM  1
N 3  14SM  1 28SM  1 42SM  1
S  126  2  3  3  7.
Each ai divides S except for ai = 12 and 28. Thus, for equation (3) to be true for all ai, G must be
even, therefore M must be odd since  ai a j is odd (each term is even except for 1  3  3, thus
the sum is odd).
The above is a one-parameter expression for a Carmichael triple. The problem now is to find odd
values of M so that the nine values of P are prime simultaneously. N will be a Carmichael
number as will N1 , N 2 , and N s , according to equation (3).
Results
The following values of M fulfilled all of the above conditions. These are the smallest values of
M that we have found. Finding them took 100 minutes on a Cray XMP computer.
M  49226306, 450936495, 533568525.
The set of primes which correspond to the smallest value of M are shown in Table 1.
Table 1. Carmichael Triple with Nine Factors
S  126 Pa  aSM  1 a  1, 2, 3; 6, 12, 18; 14, 28, 42 M  49226305
P1 
P2 
P3 
P6 
P12 
P18 
P14 
P28 
37215086581
74430173161
111645259741
86835202021
173670404041
P42 
260505606061
6202514431
12405028861
18607543291
N1  P1 P2 P3
N 2  P6 P12 P18
N 3  P14 P28 P42
N = N1 N2 N3 and has 97 digits
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CARMICHAEL NUMBERS
Table 2 details a similar Carmichael triple consisting of ten components, where N2 is of the form:
N2  6M  1 12M  1 18  1 36M  1.
Table 2. Carmichael Triple with Ten Factors
S  234 Pa  aSM  1 a  1, 2, 3; 6, 12, 18, 36; 26, 52, 78 M  806843569
188801395147
P1 
377602790293
P2 
566404185439
P3 
1132808370877
P6 
2265616741753
P12 
3398425112629
P18 
6796850225257
P36 
4908836273797
P26 
9817672547593
P52 
P78  14726508821387
N1  P1 P2 P3
N 2  P6 P12 P18 P36
N3  P26 P52 P78
N = N1 N2 N3 and has 124 digits
Following the same reasoning, one-parameter expressions for Carmichael doubles can be
derived. One very interesting Carmichael double is shown in Table 3. Note that this Carmichael
number with seven prime factors is the product of two Carmichael numbers in two different
ways!
Table 3. Carmichael Double with Seven Factors
S  36 Pa  aSM  1 a  1, 2, 3, 6; 4, 8, 12; M  735735
P1 
P2 
26486461
52972921
N1  P1 P2 P3 P6 or N1  P2 P4 P6
P3 
P6 
P4 
P8 
79459381
158918761
105945841
211891681
N 2  P4 P8 P12 or N 2  P1 P3 P8 P12
P12 
317837521
N = N1 N2 N3 and has 57 digits
Based on the previous derivation and results it is reasonable to conjecture that there are an
infinite number of Carmichael double and triples. In fact, it even seems reasonable to conjecture
that there are an infinite number of Carmichael numbers that are the product of an arbitrary
number of Charmichael numbers.
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H. DUBNER & H. NELSON
For a good introduction to Carmichael numbers, we recommend Beiler’s book [4], followed by
Ore’s book [2]. For more up to date information about Carmichael numbers and a good set of
references, see the article by Wagstaff, Jr. [3].
References:
1. J. Chernick, On Fermat’s Simple Theorem, Bulletin of the American Mathematics Society,
45, pp. 269-274, 1939.
2. O. Ore, Number Theory and Its History, McGraw-Hill, New York, p. 338, 1948.
3. S. S. Wagstaff, Jr., Large Carmichael Numbers, Mathematics Journal of Okayama
University, 22, pp. 33-41, 1980.
4. A. H. Beiler, Recreations in the Theory of Numbers, Second Edition, Dover Publications,
Inc., New York, Chapter VI, 1966.
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