Sierpinski / Riesel - Base 22

michaf · 2007-01-09, 22:22

Sierpinski / Riesel - Base 22

Conjectured Sierpinski at 6694 [5,23,97]
Conjectured Riesel at Riesel 4461 [5,23,97]

Pesky 17 k's include (now 13 to go)

Sierpinski:

22 (cedricvonck)
484 (cedricvonck)
1611 (michaf tested upto 12000)
1908 (michaf tested upto 12000)
4233 (michaf tested upto 12000)
5128 (michaf tested upto 12000)
5659 (michaf tested upto 12000)
6462 (michaf tested upto 12000)

Riesel:

1013 (michaf tested upto 12000)
2853 (michaf tested upto 12000)
3104 (michaf tested upto 12000)
3656 (michaf tested upto 12000)
4001 (michaf tested upto 12000)

22 and 484 are special cases; only non-trivials occur with n=2^m
If a prime is found for 22 case, 484 is also eliminated (n is one lower in that case)

(larger) primes found:
4118*22^12347-1 (michaf)
6234*22^16010+1 (michaf)
942*22^18359+1 (michaf)
5061*22^24048+1 (michaf)

22*22^n+1 / 484*22^n+1 status:

Code:

below (512): proven composite with phrot
(512)  22^512+1       has factor 115443366400367617
(1k)   22^1024+1      has factor 2095383775764481
(2k)   22^2048+1      has factor 65465822271579614082713282973697
(4k)   22^4096+1      has factor 40961
(8k)   22^8192+1      has factor 147457
(16k)  22^16384+1     has factor 2342241402881
(32k)  22^32768+1     has factor 65537
(64k)  22^65536+1     has factor 27918337
(128k) 22^131072+1    has factor 786433
(256k) 22^262144+1    has factor 29884417
(512k) 22^524288+1    has factor 93067411457
(1M)   no factors upto           1607651162167705601 (also P-1 stage 1 done with B1=100000 and 25 ECM curves, B1=1000, B2=100000)
(2M)   no factors upto           285159626880581633
(4M)   no factors upto           556968483053633537
(8M)   no factors upto           9221584136710389761 (stopping here)
(16M)  22^16777216+1  has factor 189162539974657
(32M)  22^33554432+1  has factor 21096518178045953
(64M)  no factor upto            9202942106325221377 (stopping here)
(128M) 22^134217728+1 has factor 91268055041
(256M) 22^268435456+1 has factor 7368180622415626241
(512M) no factor upto            9187751282130026497 (stopping here)
(1G)   no factor upto            9159662798383349761 (stopping here)

ValerieVonck · 2007-01-10, 07:24

n = 22 tested to 144895

fatphil · 2007-01-10, 16:27

Quote:

Originally Posted by CedricVonck

n = 22 tested to 144895

Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.

Xentar · 2007-01-10, 18:55

Quote:

Originally Posted by fatphil

Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.

eh.. is it the same with
18 * 18^n+1 ?
and if not, could you please try to explain it to me? (Yea, I'm trying to learn the math behind these projects

)

fatphil · 2007-01-10, 19:13

Quote:

Originally Posted by Xentar

eh.. is it the same with
18 * 18^n+1 ?
and if not, could you please try to explain it to me? (Yea, I'm trying to learn the math behind these projects

)

They are also just 18^n+1. Again, they can only be prime if n is a power of 2, and would be Generalised Fermat Numbers. Have a look at Yves Gallot's GFN page for more info.

michaf · 2007-01-10, 22:32

4118*22^12347-1 is prime
6234*22^16010+1 is prime

tests now at 18000

geoff · 2007-01-11, 05:43

If you want to sieve the GFN's, srsieve should recognise them as such and use a faster method of sieving as long as only the terms of the form n=2^m+1 appear in the input file. If you start srsieve with the -v option it will print a message something like `filtering for primes of the form (2^m)x+1' if it has recognised the sequence as GFN. (If there are non-GFN sequences in the sieve as well then it will still notice the GFN's, but most of the benefit will be lost).

ValerieVonck · 2007-01-11, 11:18

Quote:

Originally Posted by fatphil

Containing
<<<
22*22^511+1 is not prime.
22*22^2047+1 is not prime.
22*22^3583+1 is not prime.
22*22^6655+1 is not prime.
>>>

Are you sure you know what you're doing?
Those are 22^n+1, which can only ever be prime if n=2^m for some n, namely generalised Fermat numbers.

Apperantely not then... Like I said I used NewPGen to sieve the file:

Code:

k*b^n-1 with k fixed
k = 22
b = 22
n from 2 to 1.000.000

Then I fired up LLR.exe (3.7.1)

michaf · 2007-01-12, 19:23

942*22^18359+1 is prime
5061*22^24048+1 is prime

jasong · 2007-01-13, 03:05

In terms of 22^(2^m)+1, is it possible to figure the odds that at least one value of m will yield a prime for the equation? Since the numbers get big so quickly, I'm thinking it's possible this may be a Sierpinski number that may never be proven, or at least not within my lifetime.

Can anybody do the math and figure the chance that it would or wouldn't yield prime if every single m value could be figured at once for the above equation?

michaf · 2007-01-13, 09:32

The chances are quite slim I suspect,
but sieving can be done quite easily.

I have started yesterday a sieve for the first 4 numbers left, and that one did 40Mp/sec (now only 3 left since a factor was found for 512k)

It will reach the limit of srsieve in about 1 day.
If some numbers are left, well... it's getting very hard to test them.

I will start a sieve for the next 4 after the first one finishes; that sieve-speed is significantly slower (About 7Mp/sec), but will finish in reasonable time too.
After that, I think I will let it go.

2007-01-13, 03:05	#10
jasong "Jason Goatcher" Mar 2005 110110110011₂ Posts	In terms of 22^(2^m)+1, is it possible to figure the odds that at least one value of m will yield a prime for the equation? Since the numbers get big so quickly, I'm thinking it's possible this may be a Sierpinski number that may never be proven, or at least not within my lifetime. Can anybody do the math and figure the chance that it would or wouldn't yield prime if every single m value could be figured at once for the above equation? Last fiddled with by jasong on 2007-01-13 at 03:05

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2007-01-10, 22:32	#6
michaf Jan 2005 479₁₀ Posts	411822^12347-1 is prime 623422^16010+1 is prime tests now at 18000

2007-01-11, 05:43	#7
geoff Mar 2003 New Zealand 13·89 Posts	If you want to sieve the GFN's, srsieve should recognise them as such and use a faster method of sieving as long as only the terms of the form n=2^m+1 appear in the input file. If you start srsieve with the -v option it will print a message something like `filtering for primes of the form (2^m)x+1' if it has recognised the sequence as GFN. (If there are non-GFN sequences in the sieve as well then it will still notice the GFN's, but most of the benefit will be lost).

2007-01-12, 19:23	#9
michaf Jan 2005 479 Posts	94222^18359+1 is prime 506122^24048+1 is prime

2007-01-13, 09:32	#11
michaf Jan 2005 479 Posts	The chances are quite slim I suspect, but sieving can be done quite easily. I have started yesterday a sieve for the first 4 numbers left, and that one did 40Mp/sec (now only 3 left since a factor was found for 512k) It will reach the limit of srsieve in about 1 day. If some numbers are left, well... it's getting very hard to test them. I will start a sieve for the next 4 after the first one finishes; that sieve-speed is significantly slower (About 7Mp/sec), but will finish in reasonable time too. After that, I think I will let it go.