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double Mersenne number

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Mathematics Subject Classification

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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...

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

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