Exponential prime search - Page 2

Citrix · 2005-09-23, 04:14

Quote:

Originally Posted by grandpascorpion

If you don't mind sharing, what technique did you use to pare down your search?

I did random ploynomials and random seeds without sieving etc. Just a basic search to test 10 Million combinations. I found one 5 prime series and several 4 prime series. Took about a day with my program.

Citrix

grandpascorpion · 2005-09-23, 11:49

Yeah, you really need a more directed search. The polynomial is the key (preferably of the form x^2+bx+c).

I have to fiddle with this idea some more to see if there's a more effective way to sieve (once you have a good candidate polynomial).

Citrix · 2005-10-02, 05:03

a=1
b=0
c=-2

All numbers are prime when the seed is 2.

Citrix

grandpascorpion · 2005-10-02, 15:50

Yeah, all values are the SAME prime. LOL

Let's exclude cases where f(x) = x where x for some prime or some cycle of 2 is formed. Those are trivial cases.

Citrix · 2006-01-14, 04:51

l=6 a=9 b=72 c=76 seed=1
l=6 a=27 b=39 c=71 seed=4
+ several lenght 5 that I found.

I challenge everyone to find a sequence of primes with length 7 or greater? Do any such exist?

Citrix

Citrix · 2006-01-14, 22:36

More length 6.

Score 6 a4 b48 c79 seed2
Score 6 a6 b44 c39 seed2
Score 6 a45 b39 c19 seed2
Score 6 a52 b42 c87 seed2
Score 6 a64 b70 c5 seed2
Score 6 a73 b97 c23 seed2

grandpascorpion · 2006-01-17, 17:45

To improve your chances, you should keep the a term to a minimum and use a negative b. The faster the polynomial grows, the worse your odds become of finding a long sequence (if you are just using random polynomials).

How far are you going with your seed? If you good sequences at such low values, you should continue on for awhile (thru 10 million maybe). Also, I don't see why you would use a composite seed to start with. Isn't the seed the first prime in the sequence?

Unfortunately, my search went down in flames with the death of my laptop's hard drive.

Citrix · 2006-01-17, 20:54

If someone could write a fast program it would be fun to search for a series of length 7. My program is too slow to test 10M values.

Just something intresting I thought of. Using negative values for a. Starting at large numbers and using negative 'a' find the max lenght the series will go.

Citrix

grandpascorpion · 2006-01-17, 22:25

I used the tack of finding smooth polynomials but if you are randomly finding good sequences with low seeds, I would sieve out seeds that won't result in a good sequence. I'd do this for mod 30030. Start by removing 11-smooth numbers and then evaluate those remaining numbers with your function. Cull and numbers that are not 11-smooth then repeat. You would have to check for cycles of course. Unfortunately, I had a script that did this but my laptop is kaput as we speak.

Also, if you get a "hit" at such a low seed with 6, you should DEFINITELY continue forward.

A negative a won't help you either work backwards in a prime sequence. You can't "backwards" in this manner. The smaller numbers are SO cheap to check for primality (compared to the 6th or 7th terms in such a sequence) , you may as well just proceed from small to large in the sequence.

grandpascorpion · 2006-03-06, 06:11

I realized there's a much better way to search for these sequences and rewrote my code. There's still much room for improvement I think in my code.

For a=2, b=-39, c=116, there's a 7-sequence of:
3,17,31,829,1342267,3603309050281,25967672223533395319397079

I have also found a few 9 sequences but with rational (but not integral): a, b and c, one of which is:

a=1/68, b=-46/17, c=15587/68
Length=9: (7,211,313,823,7963,911173,12206891113,2191296888214271023,
70614441945552161656238786586347263

Citrix · 2006-03-06, 07:15

Cool!
What improvements did you use. If you find one with 17 primes in a row then you can get in the top 5000 list.

Citrix

2006-01-17, 17:45	#18
grandpascorpion Jan 2005 Transdniestr 503 Posts	Tweaking To improve your chances, you should keep the a term to a minimum and use a negative b. The faster the polynomial grows, the worse your odds become of finding a long sequence (if you are just using random polynomials). How far are you going with your seed? If you good sequences at such low values, you should continue on for awhile (thru 10 million maybe). Also, I don't see why you would use a composite seed to start with. Isn't the seed the first prime in the sequence? Unfortunately, my search went down in flames with the death of my laptop's hard drive.

2006-01-17, 22:25	#20
grandpascorpion Jan 2005 Transdniestr 767₈ Posts	I used the tack of finding smooth polynomials but if you are randomly finding good sequences with low seeds, I would sieve out seeds that won't result in a good sequence. I'd do this for mod 30030. Start by removing 11-smooth numbers and then evaluate those remaining numbers with your function. Cull and numbers that are not 11-smooth then repeat. You would have to check for cycles of course. Unfortunately, I had a script that did this but my laptop is kaput as we speak. Also, if you get a "hit" at such a low seed with 6, you should DEFINITELY continue forward. A negative a won't help you either work backwards in a prime sequence. You can't "backwards" in this manner. The smaller numbers are SO cheap to check for primality (compared to the 6th or 7th terms in such a sequence) , you may as well just proceed from small to large in the sequence. Last fiddled with by grandpascorpion on 2006-01-17 at 22:33

2006-03-06, 06:11	#21
grandpascorpion Jan 2005 Transdniestr 503₁₀ Posts	7-sequence found with integral a, b and c, 9-sequences found with rational a,b, and c I realized there's a much better way to search for these sequences and rewrote my code. There's still much room for improvement I think in my code. For a=2, b=-39, c=116, there's a 7-sequence of: 3,17,31,829,1342267,3603309050281,25967672223533395319397079 I have also found a few 9 sequences but with rational (but not integral): a, b and c, one of which is: a=1/68, b=-46/17, c=15587/68 Length=9: (7,211,313,823,7963,911173,12206891113,2191296888214271023, 70614441945552161656238786586347263 Last fiddled with by grandpascorpion on 2006-03-06 at 06:11

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2005-09-23, 11:49	#13
grandpascorpion Jan 2005 Transdniestr 503 Posts	Yeah, you really need a more directed search. The polynomial is the key (preferably of the form x^2+bx+c). I have to fiddle with this idea some more to see if there's a more effective way to sieve (once you have a good candidate polynomial).

2005-10-02, 05:03	#14
Citrix Jun 2003 2·7·113 Posts	a=1 b=0 c=-2 All numbers are prime when the seed is 2. Citrix

2005-10-02, 15:50	#15
grandpascorpion Jan 2005 Transdniestr 503 Posts	Yeah, all values are the SAME prime. LOL Let's exclude cases where f(x) = x where x for some prime or some cycle of 2 is formed. Those are trivial cases.

2006-01-14, 04:51	#16
Citrix Jun 2003 2·7·113 Posts	l=6 a=9 b=72 c=76 seed=1 l=6 a=27 b=39 c=71 seed=4 + several lenght 5 that I found. I challenge everyone to find a sequence of primes with length 7 or greater? Do any such exist? Citrix

2006-01-14, 22:36	#17
Citrix Jun 2003 11000101110₂ Posts	More length 6. Score 6 a4 b48 c79 seed2 Score 6 a6 b44 c39 seed2 Score 6 a45 b39 c19 seed2 Score 6 a52 b42 c87 seed2 Score 6 a64 b70 c5 seed2 Score 6 a73 b97 c23 seed2

2006-01-17, 20:54	#19
Citrix Jun 2003 2×7×113 Posts	If someone could write a fast program it would be fun to search for a series of length 7. My program is too slow to test 10M values. Just something intresting I thought of. Using negative values for a. Starting at large numbers and using negative 'a' find the max lenght the series will go. Citrix

2006-03-06, 07:15	#22
Citrix Jun 2003 2×7×113 Posts	Cool! What improvements did you use. If you find one with 17 primes in a row then you can get in the top 5000 list. Citrix