Top 3 twin found!

MooMoo2 · 2015-07-03, 18:30

4884940623*2^198800-1 is prime! (59855 decimal digits) Time : 14.689 sec.
4884940623*2^198800+1 is prime! (59855 decimal digits) Time : 14.608 sec.

https://primes.utm.edu/primes/page.php?id=120068
https://primes.utm.edu/primes/page.php?id=120069

It's the 3rd largest of all time and the largest found by an individual who's unaffiliated with any project. It also beats the original n=195000 TPS record and would have been the largest known twin if it were found before August 2009.

Stats:

Code:

Sieve start date: 		January 26, 2015
Sieve end date: 		April 8, 2015
Discovery date: 		June 30, 2015 11:45 PM
Computer used in discovery:	Intel Core i7 4790
Verification date: 		July 1, 2015 10:57 PM
Computer used in verification:	AMD Phenom II X6 1055T
Number of cores used: 		10
Total GHz: 			31.2
Sieve depth, 0-20M:		unknown
Sieve depth, 20M-4916M:		3200T
Candidates per 1M: 		306 (from 20M-4916M)
Total Primes found:		765 (includes the +1 twin)
Total Candiates tested:		approximately 1609500
Candidates tested, 0-20M:	approximately 9500
Candidates tested, 20M-4916M:	1,600,004
Riesel prime density: 		1 every 6.43M.
Candidates per riesel prime: 	2100 (from 20-4916M)
				2107 (from 0-4916M)

I was a bit lucky - the probability that a twin would be discovered after searching 4.9 G is only 29%. This is also the lowest k that produces a twin for this n.

But I wasn't that lucky. Finding it involved testing about 50% more candidates and sieving almost three times deeper than TPS's n=195000 project. Despite this, it took me less time to find the n=198800 twin than it took for TPS to find the n=195000 twin.

For those of you who're interested, the first few numbers of the decimal expansion of 4,884,940,623*2^198800+1 are 28,313,743,191,542,... ,and the last few numbers are ...2,658,872,885,249.

Batalov · 2015-07-03, 19:30

Congrats! (from someone who knows how hard it is. It is, indeed!)

Batalov · 2015-07-03, 19:37

Have you kept all intermediates? I have an offer for you that you can't refuse!

I have run on and off a certain search for which I need many approximately this sized primes. You will have 0.5 of a prime if it works, of course. PM me if interested.
I have in the past similarly contacted the Hungarians (as DavidB called them, see positions #6,7,8) and other twin searchers, as well as I've processed all of my position #5 interim half-primes.

MooMoo2 · 2015-07-03, 22:11

Quote:

Originally Posted by Batalov

Have you kept all intermediates? I have an offer for you that you can't refuse!

I have run on and off a certain search for which I need many approximately this sized primes. You will have 0.5 of a prime if it works, of course. PM me if interested.
I have in the past similarly contacted the Hungarians (as DavidB called them, see positions #6,7,8) and other twin searchers, as well as I've processed all of my position #5 interim half-primes.

Offer accepted

If anyone's interested in my lresults files and/or the raw sieve file (k=20M - 60020M, n=198800, p=3200T, about 18 million untested k/n pairs), PM me, and I'll send you the link if you'll agree to share credit for any notable primes found.

Batalov · 2015-07-13, 23:30

Unfortunately, no derivative primes were found.

Here are the examples of the tested candidates to give you a flavor of what was done:

Code:

Phi(10,(46421883*2^198800-1)*(299771367*2^198800-1)) is composite: RES64: [DB6C0836F42F050D]
Phi(5,(125526867*2^198800-1)*(299771367*2^198800-1)-1) is composite: RES64: [81248E7866F4AD15]
and a few thousand more...
There are a 200K of similar canidates easily produced and some of them are likely prime and can be found 
but the cost of sieving and testing is too high to go on for too long. 
Besides I already have a couple of primes like these.

What was of primary interest, though, and what I 100% searched were all Phi(5,p), Phi(7,p), Phi(10,p+1), Phi(14,p+1) for all of your several hundred p.
These are more interesting and rare.
They are provable by N-1 because p and p+1 are fully factored which makes
Phi(5,...)-1 and Phi(10,...)-1 50% factored and Phi(7,...)-1 and Phi(14,...)-1 33.333% factored which is enough got the proof.
Other polynomials can be tried (and tried and tried until a constructible prime is found) but they are not as interesting as Phi().
I have one of these but it was not eligible for Top5000. Just a little too small.

2015-07-03, 18:30	#1
MooMoo2 Aug 2010 1010011000₂ Posts	Top 3 twin found! 48849406232^198800-1 is prime! (59855 decimal digits) Time : 14.689 sec. 48849406232^198800+1 is prime! (59855 decimal digits) Time : 14.608 sec. https://primes.utm.edu/primes/page.php?id=120068 https://primes.utm.edu/primes/page.php?id=120069 It's the 3rd largest of all time and the largest found by an individual who's unaffiliated with any project. It also beats the original n=195000 TPS record and would have been the largest known twin if it were found before August 2009. Stats: Code: Sieve start date: January 26, 2015 Sieve end date: April 8, 2015 Discovery date: June 30, 2015 11:45 PM Computer used in discovery: Intel Core i7 4790 Verification date: July 1, 2015 10:57 PM Computer used in verification: AMD Phenom II X6 1055T Number of cores used: 10 Total GHz: 31.2 Sieve depth, 0-20M: unknown Sieve depth, 20M-4916M: 3200T Candidates per 1M: 306 (from 20M-4916M) Total Primes found: 765 (includes the +1 twin) Total Candiates tested: approximately 1609500 Candidates tested, 0-20M: approximately 9500 Candidates tested, 20M-4916M: 1,600,004 Riesel prime density: 1 every 6.43M. Candidates per riesel prime: 2100 (from 20-4916M) 2107 (from 0-4916M) I was a bit lucky - the probability that a twin would be discovered after searching 4.9 G is only 29%. This is also the lowest k that produces a twin for this n. But I wasn't that lucky. Finding it involved testing about 50% more candidates and sieving almost three times deeper than TPS's n=195000 project. Despite this, it took me less time to find the n=198800 twin than it took for TPS to find the n=195000 twin. For those of you who're interested, the first few numbers of the decimal expansion of 4,884,940,623*2^198800+1 are 28,313,743,191,542,... ,and the last few numbers are ...2,658,872,885,249.

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2015-07-03, 19:30	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2661₁₆ Posts	Congrats! (from someone who knows how hard it is. It is, indeed!)

2015-07-03, 19:37	#3
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 23141₈ Posts	Have you kept all intermediates? I have an offer for you that you can't refuse! I have run on and off a certain search for which I need many approximately this sized primes. You will have 0.5 of a prime if it works, of course. PM me if interested. I have in the past similarly contacted the Hungarians (as DavidB called them, see positions #6,7,8) and other twin searchers, as well as I've processed all of my position #5 interim half-primes.

2015-07-13, 23:30	#5
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×5²×131 Posts	Unfortunately, no derivative primes were found. Here are the examples of the tested candidates to give you a flavor of what was done: Code: Phi(10,(464218832^198800-1)(2997713672^198800-1)) is composite: RES64: [DB6C0836F42F050D] Phi(5,(1255268672^198800-1)(2997713672^198800-1)-1) is composite: RES64: [81248E7866F4AD15] and a few thousand more... There are a 200K of similar canidates easily produced and some of them are likely prime and can be found but the cost of sieving and testing is too high to go on for too long. Besides I already have a couple of primes like these. What was of primary interest, though, and what I 100% searched were all Phi(5,p), Phi(7,p), Phi(10,p+1), Phi(14,p+1) for all of your several hundred p. These are more interesting and rare. They are provable by N-1 because p and p+1 are fully factored which makes Phi(5,...)-1 and Phi(10,...)-1 50% factored and Phi(7,...)-1 and Phi(14,...)-1 33.333% factored which is enough got the proof. Other polynomials can be tried (and tried and tried until a constructible prime is found) but they are not as interesting as Phi(). I have one of these but it was not eligible for Top5000. Just a little too small.