Download MODIFIED MERSENNE NUMBERS AND PRIMES Several thousand

MODIFIED MERSENNE NUMBERS AND PRIMES
Several thousand years ago the Greek mathematician Euclid observed that the sum of two
to the nth power up to n=N equalsN
S [ N ]   2 n  2 N 1  1
n 0
This fact is easily established by summing the indicated finite geometric series. Writing
out a few terms, one hasS[1]=1+2=3
S[2]=1+2+4=7
S[3]=1+2+4+8=15
S[4]=1+2+4+8+16=31
S[5]=1+2+4+8+16+32=63
S[6]=1+2+4+8+16+32+64=127
One notices that many of these finite sums add up to prime numbers such as for N=1, 2,
4, and 6 while for others such as for N=3 and 5 produce composite numbers. The French
cleric Marin Mersenne(1588-1648) observed from these results that22-1=3, 23-1=7, 25-1=31, and 27-1=127
That is , if one takes two to a prime power p and subtracts one from the result , one will
generate a higher prime. Mersenne thought this might be true for all prime powers of
two, but his conjecture was proven wrong as already shown by 211-1=2047=23·89. The
prime powers p=13,17, and 19 work again, but p=23 and 29 fail. To date only 48 of these
so-called Mersenne Primes have been found although it is believed there are an infinite
number of them. Be that as it may, one can define a Mersenne Number as
M [ p]  2 p  1
It produces primes when p=2, 3, 5, 7, 13, 17, 19, 31,… but yields a composites when
p=11, 23, 29, 37, 41, 43, 47, … . People are still trying to find the next and 49th Mersenne
prime. It is certain to exist but lies at the border of what even the world’s largest and
fastest computers can handle.
We want here to look at a modified version of Mersenne Numbers which are guaranteed
to yield primes for any p . We do this by defining a modified Mersenne Number-


N  2 p 3k1  5k 2  7k 3  ...  1 where
k1>k2>k3>… are integers not necessarily prime
Also we require that the product term of 2p times the term in the square bracket be an
even number so that N can be an odd number. Our choice for this modification was
dictated by the fact that primes are often found at plus or minus one unit from super-
composites whose number fraction f(N) exceeds one. As we have shown earlier, these
super-composites typically have exponent vectors of the form [k1 k2 k3 ….0 0 0] with
k1>k2>k3.Typically the number of non-vanishing ks will be small. Let us demonstrate.
Consider the super-compositeS=2160=25·33·52 whose exponent vector reads [5 3 2 0 0 0]
If we look at S-1=2159 and S+1=2161, we find both of these numbers are prime. HenceN  25 [33  52 ]  1
represent two modified Mersenne Primes. Note that 25·33·52 is an even number.
We can use this approach to generate primes from non-prime Mersenne Numbers.
Consider the caseN=211[3k]-1 which is composite for k=0 but produces a prime for k=1
This prime reads N=3·211-1=6143. There are of course many other primes we can
generate by modifying the composite 211-1. For example, N=211[35·51]-1=2488319 is also
a prime.
A larger Mersenne Number is 223-1 which is composite. To make it a prime we try the
modification N=223[3k]-1 and vary k until a prime value is found. This occurs for k=10
and produces N= 495338913791 .
An even larger Mersenne Number is N=247-1=140737488355327. It factors as
N=(13264529)(2351)(4513). On multiplying 247 by the bracket [323·51] , we obtain the
large primeN=247· 323·51-1=66247372669492370111201279
This is of course not the only combination capable of producing a prime. Again note that
247· 323·51 is an even number.
What becomes clear from the above modified versions of Mersenne Numbers is that one
can always make it prime by multiplying 2p by an appropriate square bracket of the
indicated form.
Here are some other examples of modified Mersenne Primes27 [35  53 ]  1  3887999
217 [36  51 ]  1  477757439
223 [37 ]  1  18345885697
243 [311  55  72 ]  1  238599603248902963199999
It is clear that there are an infinite number of such primes and these are all easy to find. It
calls into question the ongoing efforts to find ever larger classic Mersenne Primes when
there are an infinite number of modified Mersenne Primes which can be easily generated.
January 2014