Calcprimes program

[Citrix]

I have been using the calcprimes.jar program I found on the forum to calculate the odds of finding primes. Overall I have found it very accurate. Are there any sequences that anyone knows of that beats the odds predicted by the program?

Do all the Riesel k<300 follow the odds predicted by the program? Looking at the statistics of the various drives the program looks accurate.

[pepi37]

Hi!
I just search for that program and in "my range" show 1.79 expected primes.
I should hope that will be 2 primes :))
Thanks for program, and if it is accurate as you say, that is good news for me :)

Quick sieve test ( on known set of primes)
REPORTED PREDICTED
2*10^n-1 28 23.84
3*10^n-1 19 20.45
5*10^n-1 25 24.74
6*10^n-1 31 27.63
8*10^n-1 23 23.42
9*10^n-1 20 18.05


so it looks pretty accurate to me :)

[VBCurtis]

Citrix-
A sequence that "beats the odds" is a matter of statistical distribution- there are surely a few that substantially beat the estimate so far, but we have no reason to think they will continue to do so. Like rolling dice, past performance is not connected to future results.

If you are asking if any sequence exceeds the odds and we have a reason to think it will continue to do so, that 'reason' would be publication-worthy in most cases. So, no.

Pepi-
The theory that program uses is well-known, and any inaccuracies are due to the irregular nature of the distribution of primes rather than any error/inaccuracy in the program.

[Batalov]

Originally Posted by Citrix

Do all the Riesel k<300 follow the odds predicted by the program? Looking at the statistics of the various drives the program looks accurate.

I would expect those forms that have a partial algebraic factorization to need a correction coefficient (f < 1, not in the direction of 'beating the odds'). Consider 27*2^n-1.

[Citrix]

Originally Posted by VBCurtis

If you are asking if any sequence exceeds the odds and we have a reason to think it will continue to do so, that 'reason' would be publication-worthy in most cases. So, no.

Yes that is what I was wondering. I believe (though do not have proof) that sequences that have special factors like mersenne numbers, generalized fermat, GM, GQ etc are slightly more likely than regular numbers to produce primes.

The likelihood increases in the case of mersenne numbers/GM/GQ as the size of the p increases by a factor of ln(p).

It would remain constant for a Generalized fermat series depending on the exponent. The increased likelihood for generalized fermat numbers would be ln(2^x) (where 2^x is the exponent).

Is there any way of confirming this empirically. I do not have access to the mersenne factor database/generalized fermat search database to test this.

Can anyone disprove this or give a counterexample to this?

[Mini-Geek]

Originally Posted by Batalov

I would expect those forms that have a partial algebraic factorization to need a correction coefficient (f < 1, not in the direction of 'beating the odds'). Consider 27*2^n-1.

If you've eliminated those candidates from the sieve file already, I'd expect that the program is still accurate.

[VBCurtis]

Originally Posted by Citrix

The likelihood increases in the case of mersenne numbers/GM/GQ as the size of the p increases by a factor of ln(p).

Is there any way of confirming this empirically. I do not have access to the mersenne factor database/generalized fermat search database to test this.

Can anyone disprove this or give a counterexample to this?

For Mersennes, the number of expected primes in some future range has been listed on mersenne.org for many years. The incidence of primes is too small to "prove" or "disprove" any estimate method, but I believe the rate of mersenne discovery is in rough agreement with this standard method, while your factor of ln(p) increase would estimate vastly more primes than have been discovered. At the size of exponent p currently in testing (50M+), ln(p) is over 17! That would be a massive deviation from the usual estimates.

[Citrix]

Originally Posted by VBCurtis

For Mersennes, the number of expected primes in some future range has been listed on mersenne.org for many years. The incidence of primes is too small to "prove" or "disprove" any estimate method, but I believe the rate of mersenne discovery is in rough agreement with this standard method, while your factor of ln(p) increase would estimate vastly more primes than have been discovered. At the size of exponent p currently in testing (50M+), ln(p) is over 17! That would be a massive deviation from the usual estimates.

Let me clarify:-
Odds of a regular number being prime= 1/ln(N)
Odds of a mersenne being prime (where only prime exponents are used)= [1+ln(p)]/ln(2^p-1)

For a 50M number this would be 18/50,000,000 ~ 1 in 3 million prime exponents or close to 1 prime in a 50 million n range --> which seems about right.

I agree that there is not enough data to prove or disprove this.

Odds for a GFN number being prime=[1+ln(2^x)]/ln(b^2^x)

[VBCurtis]

I think your observation is an obfuscated rephrasing of the structure of the factors of mersenne numbers, while not altering the odds any mersenne trial factored to n bits is prime using the formula applied by calcprimes.

Since mersenne factors are bigger than 2p, you can apply the "odds of prime" as if trial factoring has been done to 2p, which would increase the probability produced by the formula. However, it doesn't mean mersennes are any easier to find per primality test; it merely alters the prefactoring effort/effectiveness.

[pepi37]

Does same size of sieve(in MB) ( same base, but different K) suggested same number of primes in same range?
Why I asking this: I made test sieves until 1000000
for K 8 sieve size is 1.906 MB, and for K 96 is 1.907 MB
When using calcprimes. jar , I got predicted number of 27 primes for K=8 and only 21 primes for K=96

[Mini-Geek]

Originally Posted by pepi37

Does same size of sieve(in MB) ( same base, but different K) suggested same number of primes in same range?
Why I asking this: I made test sieves until 1000000
for K 8 sieve size is 1.906 MB, and for K 96 is 1.907 MB
When using calcprimes. jar , I got predicted number of 27 primes for K=8 and only 21 primes for K=96

Are you referring to the size of the sieved file (Newpgen or ABC format) containing the remaining candidates? I would only expect this if the K=96 file took up significantly more bytes per candidate.