> For these reasons (and for the fact that among the only other class of
> numbers known to admit DWT arithmetic, the Fermats, there are vastly fewer
> primality candidates than among the Mersennes), I predict that a Mersenne
> prime will hold the record for a long time to come.
Not true - DWT can be applied to any polynomial of the form x^n+/-1.
- If n > 1 and x^n-1 is prime, then x=2 and n is prime. With the form x^n-1,
only the Mersenne numbers can be tested.
- If x > 2 and x^n+1 is prime, then x is even and n=2^m. With the form
x^n+1, a large set of numbers can be tested the Generalized Fermat numbers.
In a fixed range, they are more numerous than Mersenne numbers. Because
n=2^m, we only have to use a FFT of length a power of 2. We have to use a
base representation which is not a power of 2. If we consider that the
advantage cancels the drawback, the test of a GFN is about as fast as a test
of a Mersenne number.
> There are two further reasons why non-DWT classes of numbers will be less
> likely to yield a world-record-sized prime:
>
> (1) volunteers for distributed computing efforts will always tend to find
> the prospect of findng a world-record size something more interesting
> than something arcane like 'the largest known irregular Huffergnu"ten
> prime octuplet,' i.e. GIMPS will attract more compute cycles because
> it holds the current prime record, and by a whopping margin; and
True.
> (2) Among the Proths, the fact there are so many possible kinds of
> candidates dilutes the effort, i.e. makes it vastly more
> time-consuming to test all the candidates up below any
> reasonably-sized threshold.
The major advantage of the other forms is the fact that you don't have to
organize the search! Because if the number of candidates is huge, the
probability for a number to be tested twice is very small. The Mersenne
number form has only one degree of freedom. If some forms with more degrees
of freedom are as fast to test, the volunteers also will be more free.
Yves
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