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Homedouble Mersenne number

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# double Mersenne number

A double Mersenne number is a number of the form $2^{{2^{p}-1}}-1$. Put another way, negative one plus 2 raised to the power of a Mersenne number. The first few double Mersenne numbers (and the ones small enough to show here) are: 3, 7, 127, 170141183460469231731687303715884105727.

If a double Mersenne number is itself prime, then it is called a double Mersenne prime. Obviously its index is then a Mersenne prime. The four double Mersenne numbers listed above are all primes, but as of today, $2^{{170141183460469231731687303715884105727}}-1$ is a probable prime, despite an intense effort to find factors. (According to the Prime Pages, if it’s composite, its least prime factor must be at least $5\times 10^{{51}}$.

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## Comments

## Catalan-Mersenne, C5, is composite

Here's a brief, but sustainable comment...

The next Catalan number, C5, is composite.

A number of this size is usually proven prime by raising it ex-

ponentially to a carefully-selected base with the expectation of

discovering a specific residue after modulation.

or... a^(modulendum-1)== residue(modulator) iff (some criteria).

However, double-Mersenne numbers not only describe their partic-

ular format, they also self-indicate their primalities when the

correct (modulator) is chosen.

Begin with M(M(p+1)) such that 'p' is prime and carefully calc-

ulate their self-predicting (modulator) as 2^((p+1) -1 -1) -1 -1

or... 2^(p-1)-2.

Now, simply walk through the double-Mersennes, until you reach the

Catalan numbers, using this bit of information to discover that

all prime Catalan numbers are linked to their 'residue' formula...

2^(p-2)-1.

C1 is not testable, but is nevertheless prime, and C2, C3, and C4

& all double-Mprimes have to identify precisely w/the statement:

[M(M(p+1))= 2^(2^(p+1)-1)-1] == 2^(p-2)-1 (mod (2^(p-1)-2)).

I verified that M(M(128))== 2^80-1 {<> [2^125-1]} (mod 2^126-2)

using the GNU/Bignum {Try GMP!} interpreter from their website.

The next Catalan number, C5, is composite due to this result.

Further manipulation of the above statement also predicts the

nature of the 'p' candidates...

Just remove the exponents of the (modulendum) and (modulator) to

arrive at both 2^(p+1)-1 and p-1, respectively; and we only need

to compare 2^(p+1) versus 'p' to reveal their connection:

2^(p+1)-2== 2(mod p)... which is equivalent to the 2-PRP test.

Later, it would be more accurately discovered that only prime num-

bers can contribute to the production of the double-Mersenne num-

bers that we call Catalan numbers.

These two congruencies provide different but natural methods for

predicting the primality of double-Merennes -- Catalan numbers --

due to their format; check a few other double-Mersenne numbers if

you like...

Bill Bouris, Aurora, IL USA