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Mersenne Composites and Cyclotomic Primes
Jim MacDougall
School of Mathematical & Physical Sciences
University of Newcastle
N.S.W. 2308
Australia
One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne
primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the
form 2p − 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians’ quest for
the so-called perfect numbers, those like 6 and 28 which are the sum of their
proper divisors. The connection was given in Euclid’s Elements in 300 B.C.:
if 2p − 1 is prime, then 2p−1 (2p − 1) is a perfect number. Much later, Euler
proved that all the even perfect numbers correspond to Mersenne primes. So
the interest for many years has been in finding Mersenne primes. Only 39 are
known, including several monsters discovered in recent years using thousands
of PC’s coordinated via the Internet (see [1] for information on the project).
Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of
them.
Perhaps the Mersenne primes have had more than their fair share of
publicity and our focus should be expanded to include a family of composite
numbers which enjoy a similar algebraic property. Any number of the form
xn − 1 can be factored algebraically:
xn − 1 = (x − 1)(xn−1 + · · · + 1)
and clearly such a number can be prime (hence a Mersenne prime) only when
both x = 2 and the factor 2n−1 + · · · + 1 happens to be prime. I would like
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to widen the perspective a bit - I think a number like 26 = 33 − 1 has equal
merit because when x3 − 1 is factored completely,
x3 − 1 = (x − 1)(x2 + x + 1)
26 = (3 − 1)(32 + 3 + 1)
= 2 · 13
both algebraic factors yield primes. Similarly, when x9 − 1 is factored completely, we have
x9 − 1 = (x − 1)(x2 + x + 1)(x6 + x3 + 1)
and
19682 = 39 − 1 = (3 − 1)(32 + 3 + 1)(36 + 33 + 1) = 2 · 13 · 757
In other words, the prime factorisation of the number agrees with the algebraic factorisation. In the case of Mersenne primes, the fact that x − 1
happens to be 1 (and not a prime), leaving us with single prime factor for
the number, is just an accident of nature and should not merit the undue
attention it gets.
The expression xn − 1 can be factored algebraically into the cyclotomic
polynomials and the examples above are typical. In fact we know (e.g., see
[2], p. lxvii) that
Y
Φd (x)
xn − 1 =
d|n
where the Φd (x) are the cyclotomic polynomials of degree d, and are known
to be irreducible over the integers. In the examples above, none of the
cyclotomic factors can be further factored numerically after substituting 3
for x. As a contrast, the (irreducible) cyclotomic factorisation of x5 − 1 is
(x − 1) (x4 + x3 + x2 + x + 1) while substitution of x = 3 gives
¡
¢
35 − 1 = (3 − 1) 34 + 33 + 32 + 3 + 1
= 2.11.11
and the integer factorisation does not agree with the polynomial factorisation.
So I propose that we call any number of the form whose prime factorisation agrees with its cyclotomic polynomial factorisation a Mersenne prime
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or Mersenne composite, as the case may be, and the numerical factors cyclotomic primes. Here are a few more examples of Mersenne composites:
46655 = 66 − 1 = (6 − 1)(6 + 1)(62 − 6 + 1)(62 + 6 + 1) = 5 · 7 · 31 · 43
65535 = 48 − 1 = (4 − 1)(4 + 1)(42 + 1)(44 + 1) = 3 · 5 · 17 · 257
From the examples we see that 2, 3, 5, 7, 11, 13, 17, 31, 43, 257 and 757
are all cyclotomic primes. To determine which other primes are cyclotomic
primes we would have to have a characterisation of Mersenne composites. It
may be that every prime is cyclotomic.
Let us note in passing that for certain bases there are other algebraic
factorisations of numbers of the form xn − 1, the Aurifeuillian factorisations
(see [2]). I am concerned only with the cyclotomic factorisations described
above.
Under what conditions can a number an − 1 be a Mersenne composite?
Here are some fairly obvious necessary conditions on a and n:
• Because a − 1 is always a factor of an − 1, we must have a = 2 or
a = p + 1 for some prime p.
• If n is even (and a > 2), then a + 1 is also a factor of an − 1, so must be
prime also. That is, a must lie between twin primes. In other words, if
a does not lie between twin primes, then n must be odd.
• If a lies between twin primes then a2 −1 is always a Mersenne composite.
We don’t know if there are infinitely many such composites, since it’s
not known if there are infinitely many twin primes.
• If p is any prime dividing n, then ap −1 is a factor of an −1 and so an −1
will not have the Mersenne property unless ap − 1 does. Thus when
a = 2, ap − 1 must be a Mersenne prime. So n can contain only prime
factors that yield Mersenne primes. (This is not a sufficient condition,
however. For example, 221 −1 is not a Mersenne composite even though
23 − 1 and 27 − 1 are Mersenne primes.)
Trying to find further restrictions reveals a number of interesting subproblems. For example, if n is a multiple of 3, then an − 1 is divisible by
a3 − 1 and hence by a2 + a + 1. For which a = p + 1 is a2 + a + 1 a prime? In
other words, for what primes is p2 + 3p + 3 a prime? As far as I know, no one
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knows the answer. The more general question arises for any prime divisor
q−1
P i
q of n: we need to know when the expression
a is a prime. Similarly,
0
when n is a multiple of 6, we need p2 + p + 1 to be prime. Again, I don’t
think anyone knows when this is true. Other exponents yield other similar
problems.
Another interesting instance arises when a = 2 and n is a power of 2;
then all the factors are of similar form. For example,
216 − 1 = (2 + 1)(22 + 1)(24 + 1)(28 + 1).
k
These factors of the form 22 +1 are the well-known Fermat numbers, thought
by Fermat to be prime for any positive exponent k. It is now known that
Fermat was wrong: 232 + 1 has been known to be composite for a long time
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(in fact, no subsequent Fermat numbers are known to be prime). Since 22 +1
k
is composite, 22 − 1 does not have the Mersenne composite property for any
k > 5.
Some infinite sets of values for a and n can be ruled out. For example, if
n (bigger than 4) is a multiple of 4, then it is not hard to see that a must be
a multiple of 10. The reader can also show that if a = 3, then n can’t be a
multiple of 5.
Using some of the above ideas, I started a computer search for Mersenne
composites using Maple on my PC to factor the numbers an − 1. In particular, I first checked the prime values of the exponent n. Then I only needed
to check those composite exponents all of whose prime factors already gave
Mersenne primes or composites. After some initial success, I was soon asking
Maple to factor numbers that were bigger than it could handle in a reasonable
amount of time. It was then that I remembered the Cunningham Project.
This project which started in 1925 has been dedicated to factoring numbers
precisely of the form an ± 1. Tables of factors for 2 ≤ a ≤ 12 have been
published in book form [2] in 1988. The project is being continually updated
with now exponents treated over as wide a range as modern factorisation
techniques permit. The most recent tables are available on-line at [3]. Using the Cunningham tables helped me more quickly pick out the Mersenne
composites.
As I proceeded to compile the entries listed in the tables, I realized that
I was wasting effort. I didn’t need to factor the large numbers involved,
only decide if they were prime. Maple has a function which tests whether
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Prime Exponents
Composite Exponents
2, 3, 5, 7, 13, 17, 19, 31, 61, 4, 6, 8, 9, 10, 12, 14, 15, 16, 24, 26, 27,
89, 107, 127, 521, 607
30, 32, 34, 38, 49, 62, 65, 85, 93, 98,
122, 133, 170, 254
Table 1: Exponents for Mersenne Primes and Composites with Base 2
a
3
4
6
8
12
14
18
20
24
30
32
n
< 800
< 600
< 600
< 500
< 400
< 400
< 350
< 350
< 350
< 300
< 300
Prime Exponents
3, 7, 13, 71, 103, 541
2
2, 3, 7, 29, 71, 127, 271, 509
3
2, 3, 5, 19, 97, 109, 317, 353
3, 7, 19, 31, 41
2
3, 11, 17
3, 5, 19, 53, 71
2, 5, 11, 163
−−
Composite Exponents
9, 21, 63
4, 8, 16
4, 6, 8, 21
9
10, 218
−−
−−
9
−−
−−
−−
Table 2: Mersenne Composites with Other Bases
a number is prime without trying to find its factors (which is usually a far
more difficult operation) and I thought this would make the task much easier.
Table 1 lists the exponents n < 1200 corresponding to the base a = 2. The
prime exponents, of course, give the usual Mersenne primes and there are 14
of them in this range. There are 25 composite exponents which also yield
numbers with the cyclotomic factorisation property. Unfortunately, Maple’s
primality function relies on pseudoprime tests and so while completely correct
for smaller numbers (up to about 14 digits) is not an absolute verification
of primality for larger numbers. So I turned to the marvellous computer
algebra system Magma which has an implementation of the elliptic curve
prime-proving method. Magma confirmed the results of Table 1.
The second table lists exponents for all possible bases a ≤ 32 which can
yield Mersenne composites. The range of values of n depends on the base the largest numbers tested for each base have in the order of 400 digits.
The numbers displayed in the tables invite lots of observations and questions. There appear to be many more Mersenne numbers to base 2 than to
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any larger base. Whether this would be true if we tested a much larger range
of exponents remains to be seen. It is especially interesting, though, how few
there are with base larger than 2 and composite exponents. In particular,
is there just a single one with base 8 or base 20? Are 2, 4, 8, 16 the only
exponents for base 4? (and why are they only powers of 2?) In view of the
fact that there are none in the range I checked with composite exponents for
bases 14, 18, 24, and 30, could it be that there are none at all for these (and
some other bases)?
Realistically, we are still only dealing with small numbers, so it is dangerous to conjecture too much on this small amount of evidence. I think
the important question is whether there is only a finite number of these
altogether. Given the apparently sporadic distribution of the traditional
Mersenne primes, one should probably also expect that for Mersenne composites.
References
[1] http://www.mersenne.org/
[2] Factorizations of bn ± 1, b = 2,3,5,6,7,10,11,12 up to High Powers, Brillhart, J., Lehmer, D., Selfridge, J.L., Tuckerman, B., Wagstaff, S.; Amer.
Math. Soc., 1988
[3] ftp://sable.ox.ac.uk/pub/math/cunningham/
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