532 Prime Grid

[spkarra]

I started gathering Primes that fall under the 532 Prime Grid. viz.
5*(3^n) + OR - 2.
For example:
3^1=3; Mult with 5 = 15; Plus 2 = 17 (Length = 2) --- This is a Prime.

As of today, the last one that I found was:
3^5750 Multiply with 5 Minus 2 (Length = 2745) --- is a Prime.

My target is to find the first Prime in this Grid that is 100,000 digits long.

[kar_bon]

100,000 digits for 5*3^n+/-2 is at about n=209600!

The problem should be to proove such number prime!

[gd_barnes]

Originally Posted by spkarra

I started gathering Primes that fall under the 532 Prime Grid. viz.
5*(3^n) + OR - 2.
For example:
3^1=3; Mult with 5 = 15; Plus 2 = 17 (Length = 2) --- This is a Prime.

As of today, the last one that I found was:
3^5750 Multiply with 5 Minus 2 (Length = 2745) --- is a Prime.

My target is to find the first Prime in this Grid that is 100,000 digits long.

It is virtually impossible to prove a 100,000-digit number of the form 5*3^n+2 or 5*3^n-2 is prime with today's mathematical knowledge and technology. Only large numbers where p+1 or p-1 can be fully factored to at least 33% can be proven prime, i.e. k*2^n-1, k*3^n-1, k*2^n+1, k*3^n+1, etc.

You can use the software PRIMO to prove that numbers of your form < ~15,000 digits are prime but it can take quite a while. (Anything > ~10,000 digits takes months to prove.) For larger than that, all that you can hope to do is show that your form is a PRP or probable prime. Nevertheless, you could easily be the first to find a PRP of such a form that is greater than only 10,000 digits. Such PRPs are shown at the PRP top records site at http://www.primenumbers.net/prptop/prptop.php. There are no PRPs with those forms there yet.

As a point of reference though on the amount of CPU time that is involved, the n-value for a 10,000-digit PRP would need to be n>=20,960. Such a test should take about 13 times longer than your test for your prime at n=5750 and 48 times longer to find a prime! Further...a test for a 100,000-digit PRP at n>=209,600 should take 1,300 times longer than your test for your prime at n=5750 and 48,000 times longer to find a prime!! The testing time scales with the square of the n-value and the amount of time to find a prime scales with the cube of the n-value.

To find a 100,000-digit PRP, I would suggest at least a single quad-core machine running all 4 cores 24 hours a day 7 days a week. Even then, it's possible that it could take you several months to find one. Be sure and do an adequate amount of sieving first or you will end up taking much more time than is necessary.

I just wanted you to be aware of the amount of effort involved. We'll be glad to help you analyze the n-range that you are considering testing and recommend the best way of sieving and testing.


Gary

[spkarra]

Originally Posted by kar_bon

100,000 digits for 5*3^n+/-2 is at about n=209600!

The problem should be to proove such number prime!

Thanks a lot....Its becoming more challenging....

[gd_barnes]

Originally Posted by spkarra

Thanks a lot....Its becoming more challenging....

What software are you using to prove the numbers prime? There's a big difference between finding PRPs with LLR, PFGW, Prime95, etc. and actually proving them prime with PRIMO or other similar software. At 10,000 digits, it will likely take several CPU months to prove a number prime of the form that you are searching. 10,000 digits will be reached at just a little above n=20,000.

[spkarra]

I found out that when n <10,000 the following n values belong to 532 Prime Grid.
*************************

5*3^1 Plus 2 ; Length ; 2
5*3^1 Minus 2 ; Length ; 2
5*3^2 Plus 2 ; Length ; 2
5*3^2 Minus 2 ; Length ; 2
5*3^3 Plus 2 ; Length ; 3
5*3^5 Plus 2 ; Length ; 4
5*3^5 Minus 2 ; Length ; 4
5*3^6 Minus 2 ; Length ; 4
5*3^7 Plus 2 ; Length ; 5
5*3^8 Minus 2 ; Length ; 5
5*3^10 Plus 2 ; Length ; 6
5*3^11 Plus 2 ; Length ; 6
5*3^11 Minus 2 ; Length ; 6
5*3^12 Minus 2 ; Length ; 7
5*3^13 Plus 2 ; Length ; 7
5*3^14 Minus 2 ; Length ; 8
5*3^16 Plus 2 ; Length ; 9
5*3^17 Minus 2 ; Length ; 9
5*3^26 Minus 2 ; Length ; 14
5*3^28 Plus 2 ; Length ; 15
5*3^32 Plus 2 ; Length ; 16
5*3^47 Minus 2 ;  Length ; 24
5*3^56 Plus 2 ;   Length ; 28
5*3^57 Plus 2 ;   Length ; 28
5*3^59 Minus 2 ;  Length ; 29
5*3^62 Plus 2  ;  Length ; 31
5*3^65 Minus 2 ;  Length ; 32
5*3^83 Minus 2 ;  Length ; 41
5*3^92 Minus 2 ;  Length ; 45
5*3^95 Plus 2 ;   Length ; 47
5*3^96   Minus 2 ;  Length ; 47
5*3^110  Minus 2 ;  Length ; 54
5*3^111  Plus 2 ;   Length ; 54
5*3^113  Minus 2 ;  Length ; 55
5*3^160  Plus 2 ;   Length ; 78
5*3^186  Minus 2 ;  Length ; 90
5*3^308  Plus 2 ;   Length ; 148
5*3^323  Plus 2 ;   Length ; 155
5*3^527  Minus 2 ;  Length ; 253
5*3^834  Minus 2 ;  Length ; 399
5*3^855  Plus 2 ;   Length ; 409
5*3^880  Plus 2 ;   Length ; 421
5*3^1081  Plus 2 ;  Length ; 517
5*3^1095  Plus 2 ;  Length ; 524
5*3^1274  Minus 2 ; Length ; 609
5*3^1288  Plus 2 ;  Length ; 616
5*3^2635  Plus 2 ;  Length ; 1258
5*3^2822  Plus 2 ;  Length ; 1348
5*3^2948  Plus 2 ;  Length ; 1408
5*3^3195  Plus 2 ;  Length ; 1526
5*3^5124  Minus 2 ;  Length ; 2446
5*3^5750  Minus 2 ;  Length ; 2745
5*3^7616  Minus 2 ;  Length ; 3635
5*3^7702  Plus 2 ;   Length ; 3676
5*3^8823  Plus 2 ;   Length ; 4211

****************************

[gd_barnes]

Have you proven them all prime?

[spkarra]

Originally Posted by gd_barnes

Have you proven them all prime?

Yes! I used JAVA's builtin fuction IsPrime. I will be glad to provide the BigIntegers, if you or anyone wants to cross-check.

[Mini-Geek]

Originally Posted by spkarra

Yes! I used JAVA's builtin fuction IsPrime. I will be glad to provide the BigIntegers, if you or anyone wants to cross-check.

I don't know of a built-in function IsPrime that proves primality. There's an isProbablePrime under BigInteger, but that's just PRP, not proven. Can you please show us the code (or at least part of it, e.g. a description explaining it) you used to 'prove' these primes.
Java isn't the fastest way to prove these numbers, and the larger ones (unless I'm missing a faster option than ECPP with Primo) will take on the order of hours to days to prove.

[Batalov]

The 10,000-digit boundary will look like this:
...
5*3^20445+2 ( 9756 digits)
5*3^26863+2 (12818 digits)
5*3^28501+2 (13600 digits)
5*3^29086+2 (13879 digits)
5*3^30926+2 (14757 digits)
and no "-2" primes for miles and miles and miles..!
...

[gd_barnes]

Originally Posted by spkarra

Yes! I used JAVA's builtin fuction IsPrime. I will be glad to provide the BigIntegers, if you or anyone wants to cross-check.

Spkarra,

Based on Tim's (Mini-Geek) comments, can you provide details, BigIntegers, or whatever else you have that proves the primes. Like he said, it is a multi-hour task to prove some of the larger PRPs on your list.

For your future reference, can you refer to your primes in their "normal" form as in:
5*3^7702+2 instead of 5*3^7702 Plus 2
5*3^7616-2 instead of 5*3^7616 Minus 2

It makes them easier to read. Thanks.


Just to let you know, Karsten (kar_bon), one of our co-admins, has corrected the typos that you pointed out in your original list and deleted your subsequent posting. That will make it easier for historical reference.


Gary