Wheel factorization: Efficient?

[CRGreathouse]

Originally Posted by science_man_88

no I have an idea just can't prove it. really all we have to do is prove if 6np+/-p can ever intercept 4n+1

My interpretation:

"For a given p, can we determine if there exist positive integers m,n with ?"

Originally Posted by science_man_88

and is that n a 4n+1 in the sequence

My interpretation:

"and such that there exists a positive integer a such that "

Originally Posted by science_man_88

if so the number is eliminated. but I wanted to use indexes.

So we're back to post #359: you're trial-dividing Mersenne numbers by (4^a-1)/3.

[science_man_88]

Originally Posted by CRGreathouse

My interpretation:

"For a given p, can we determine if there exist positive integers m,n with ?"



My interpretation:

"and such that there exists a positive integer a such that "



So we're back to post #359: you're trial-dividing Mersenne numbers by (4^a-1)/3.

really because I'm coming up with equations with a remainder hence I know they won't divide the Mersenne number.

[CRGreathouse]

Originally Posted by science_man_88

is this the polite way of calling me an idiot if so I like it, hurts me a lot less.

It's my way of saying that I don't think you have a method -- you're just hoping to develop a method.

If you'd like me to judge your intelligence, I bill at $120/hour (min. 3 hours) plus reasonable expenses.

Originally Posted by science_man_88

look can you tell when 4x+1 = 6np+/-p ?

What's given and what are we finding? If, as in my formulation above, p is given and x and n are to be determined, this is simple.

Originally Posted by science_man_88

if so is there a way to relate p with x so knowing a p we can prove if a x exist for that p in a quick manner if we can have a quick true false test for this we can figure this out.

Maybe you should write out your test (possibly in Pari) on the assumption that x and n have been found.

[CRGreathouse]

Originally Posted by science_man_88

really because I'm coming up with equations with a remainder hence I know they won't divide the Mersenne number.

What equations would those be?

[science_man_88]

Originally Posted by CRGreathouse

What equations would those be?

I was thinking Mersenne composite / m = 24 remainder 7

I have no idea how to figure it out anymore. obviously I have no reason to post.

[CRGreathouse]

Originally Posted by science_man_88

I have no idea how to figure it out anymore. obviously I have no reason to post.

...says the person with 100 posts in the last week.

Think over it for a while, collect your thoughts, and post again when you have a good idea of what you want to do.

[3.14159]

@CRG: For the prime search, I've done sieving up to 5,090,000,000,000. I was hoping that a few more candidates would be kicked, but NewPGen only kicked about 110 more candidates.

Recommended sieve params (NewPGen: limit = 1152921504606846976):

Special-form primes in between:
1-1500 digits: Waste no effort sieving for this range, Proth.exe will take care of this. (If they are Generalized Proth, that is.) (Else, use PrimeForm.) Ex: 7000 * 89³⁶⁰ + 1
1500-10k digits: Sieve to 10^11
10k - 25k digits: Sieve to 10^12
25k + digits: Sieve to ≥ 10^12.

[science_man_88]

by the way CRGreathouse according to the forum I'm only 6-7 post behind you this week.

first one useful fits 4x+1

second one fits 4x+1 and all the ones after and including it fit 16x+5 we can predict when these will intercept 6np+/-p no doubt can we then find a way to eliminate them assuming they be equal for more than one equation is there a pattern to it if so eliminate the ones that fit it. then repeat for other p up to a limit. oh and if we can get an efficiency out of this a 32 bit cpu should be able to use those patterns to sift through all odd Mersenne numbers up to 2^(2^33+3)-1 without bit extension. and a 64 bit would be able to go to 2^(2^65+3)-1 without extension i think. so I really would want this to work think this would be kinda cool to find a prime without factoring if possible.

[3.14159]

Now I realized why base 12096 yielded almost nothing: Too many factors. I'm going to go back to 1297, which is prime, in addition to the base 2 and base 113 searches I'm performing.

So far, I got an instant 1 in 18. Sieving progress on the other two searches:

k * 2²⁵⁶⁷²⁰ + 1 -> 6 * 10¹²
k * 113²⁸⁷²⁰ + 1 -> 1.4 * 10¹²

[3.14159]

Originally Posted by science_man_88

so I really would want this to work think this would be kinda cool to find a prime without factoring if possible.

Primality tests do not need any factoring to work. Albeit trial factoring is used to eliminate obvious composites.

Also:

Originally Posted by science_man_88

first one useful fits 4x+1

second one fits 4x+1 and all the ones after and including it fit 16x+5 we can predict when these will intercept 6np+/-p no doubt can we then find a way to eliminate them assuming they be equal for more than one equation is there a pattern to it if so eliminate the ones that fit it. then repeat for other p up to a limit.

This is too vague for me. Many things are left unspecified here.

[CRGreathouse]

Originally Posted by 3.14159

Now I realized why base 12096 yielded almost nothing: Too many factors.

?

The more (small distinct) prime factors in the base, the better. A base of 12096 'protects' you from 2, 3, and 7; you find 3.5 times the expected number of primes vs. random numbers of the same size. For base 1297 you find 1.0008 times the expected number of primes.

Primorial bases are, naturally, the best for a given size. I imagine they're largely taken? 30030, for example, would produce 5.2135 times the normal number of primes.