Some time ago the question of whether any of the Mersenne primes are
twins came up, and I recently decided to investigate the question in
some depth and share my results.
First, we consider the case that a Mersenne prime is the lesser member
of a pair of twin primes. Two results from number theory are useful
here: If 2^n-1 is prime, then n is prime. And if 2^n+1 is prime, then n
is a power of two. Thus 2^n-1 and 2^n+1 are both prime only when n is
simultaneously prime and a power of two, that is, when n=2. And so 2^2-1
is the only example in this case.
Second, we look for cases where a Mersenne prime is the greater of two
twin primes. Here we must test numbers of the form 2^p-3 where p an
exponent of a known Mersenne prime. The exponent 2 is too small. The
exponents 3 and 5 yield the primes 2^3-3 and 2^5-3. In each of the 34
remaining cases, 2^p-3 is composite.
To be completely sure of this, I found as many first factors as possible
for numbers of this form.
p First factor of 2^p-3
----------------------------------------
3* 5
5* 29
7 5
13 19
17 53
19 5
31 5
61 29
89 29
107 5
127 5
521 5296777
607 5
1279 5
2203 5
2281 19
3217 19
4253 No factor to 2^32
4423 5
9689 53
9941 23
11213 No factor to 2^32
19937 47
21701 53
23209 53
44497 29
86243 5
110503 5
132049 17040433
216091 5
756839 5
859433 4219
1257787 5
1398269 29
2976221 293
3021377 10513
*Indicates that 2^p-3 is prime.
As you can see, there are only two numbers (2^4253-3 and 2^11213-3) in
the table that I have not been able to factor. Both fail the probable
prime test to base 2. Both of these numbers appear to be worthy
challenges for any factoring effort. If someone knows of any factors of
these numbers or an easy way to find them, then I'd be interested in
hearing about it. Are there any rules for these numbers analagous to the
rules for prime factors of Mersenne numbers?
For the largest exponents in the table, it is very fortunate that the
numbers in question happen to have small factors, for otherwise even a
non-constructive proof of compositeness would probably be impossible at
present.
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