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-   -   4052555153018976267*2^n+1 (Pierpont 3^39*2^n+1) (https://www.mersenneforum.org/showthread.php?t=27202)

 Alex 2021-10-09 15:34

4052555153018976267*2^n+1 (Pierpont 3^39*2^n+1)

Hello.
I`m working on [B]4052555153018976267*2^n+1[/B] (Pierpont [B]3^39*2^n+1[/B]) with [B]n=[1..10.000.000][/B].
The range was sieved up to p=114e12.
I use Sr1sieve, PFGW, LLR.

Now it is checked n=[1..1.000.000] (49.127 of 491.804 tests).

34 small primes found:

[CODE]4052555153018976267*2^2+1 is prime! (20 decimal digits) Time : 1.235 ms.
4052555153018976267*2^11+1 is prime! (22 decimal digits) Time : 2.454 ms.
4052555153018976267*2^82+1 is prime! (44 decimal digits) Time : 16.804 ms.
4052555153018976267*2^100+1 is prime! (49 decimal digits) Time : 28.723 ms.
4052555153018976267*2^106+1 is prime! (51 decimal digits) Time : 29.094 ms.
4052555153018976267*2^131+1 is prime! (59 decimal digits) Time : 48.247 ms.
4052555153018976267*2^160+1 is prime! (67 decimal digits) Time : 69.903 ms.
4052555153018976267*2^310+1 is prime! (112 decimal digits) Time : 503.097 ms.
4052555153018976267*2^490+1 is prime! (167 decimal digits) Time : 2.631 sec.
4052555153018976267*2^1052+1 is prime! (336 decimal digits) Time : 118.200 ms.
4052555153018976267*2^1486+1 is prime! (466 decimal digits) Time : 44.392 ms.
4052555153018976267*2^1532+1 is prime! (480 decimal digits) Time : 48.573 ms.
4052555153018976267*2^2995+1 is prime! (921 decimal digits) Time : 87.253 ms.
4052555153018976267*2^4295+1 is prime! (1312 decimal digits) Time : 67.959 ms.
4052555153018976267*2^8542+1 is prime! (2591 decimal digits) Time : 142.527 ms.
4052555153018976267*2^10531+1 is prime! (3189 decimal digits) Time : 183.830 ms.
4052555153018976267*2^11522+1 is prime! (3488 decimal digits) Time : 186.658 ms.
4052555153018976267*2^19975+1 is prime! (6032 decimal digits) Time : 465.413 ms.
4052555153018976267*2^23587+1 is prime! (7120 decimal digits) Time : 689.028 ms.
4052555153018976267*2^25202+1 is prime! (7606 decimal digits) Time : 751.311 ms.
4052555153018976267*2^26938+1 is prime! (8128 decimal digits) Time : 779.696 ms.
4052555153018976267*2^28448+1 is prime! (8583 decimal digits) Time : 929.404 ms.
4052555153018976267*2^30980+1 is prime! (9345 decimal digits) Time : 1.017 sec.
4052555153018976267*2^48527+1 is prime! (14627 decimal digits) Time : 2.692 sec.
4052555153018976267*2^64607+1 is prime! (19468 decimal digits) Time : 4.952 sec.
4052555153018976267*2^251027+1 is prime! (75586 decimal digits) Time : 81.627 sec.
4052555153018976267*2^283288+1 is prime! (85297 decimal digits) Time : 114.207 sec.
4052555153018976267*2^313411+1 is prime! (94365 decimal digits) Time : 126.248 sec.
4052555153018976267*2^374870+1 is prime! (112866 decimal digits) Time : 199.486 sec.
4052555153018976267*2^446486+1 is prime! (134425 decimal digits) Time : 303.894 sec.
4052555153018976267*2^513392+1 is prime! (154565 decimal digits) Time : 386.964 sec.
4052555153018976267*2^648871+1 is prime! (195349 decimal digits) Time : 570.565 sec.
4052555153018976267*2^759071+1 is prime! (228522 decimal digits) Time : 844.726 sec.
4052555153018976267*2^903875+1 is prime! (272113 decimal digits) Time : 1218.959 sec.[/CODE]

 bur 2021-10-11 07:14

Interesting, I didn't know about Pierpont primes. Did you check for ((x)G))F divisibility? The chances are slim, but you never know...

 Alex 2021-10-14 09:27

Hello, bur.

I myself accidentally found out about Pierpont primes looking someone's old results page.

3^39 is max < 2^63.

When I discovered a new Proth prime (P), I check a lot of possible chains and numbers near it:

[CODE]F() & GF() & xGF() // Possible Fermat divisor
P+2 // Possible Twins
P-2 // Possible Twins
P+4 // Possible Cousins
P-4 // Possible Cousins
P+6 // Possible Sexy or part of AP
P-6 // Possible Sexy or part of AP
P+8 // Possible part of tuplet
P-8 // Possible part of tuplet
P+10 // Possible part of tuplet
P-10 // Possible part of tuplet
P+12 // Possible part of AP
P-12 // Possible part of AP
P+18 // Possible part of AP
P-18 // Possible part of AP
P+24 // Possible part of AP
P-24 // Possible part of AP
P+30 // Possible part of AP
P-30 // Possible part of AP
P+210 // Possible part of AP
P-210 // Possible part of AP
2*P+1 // Possible Cunningham 1st or Safe / Sophie Germain
2*P-1 // Possible Cunningham 2nd
(P+1)/2 // Possible Cunningham 2nd[/CODE]

 dannyridel 2021-10-14 15:21

[QUOTE=Alex;590022]Hello.
I`m working on [B]4052555153018976267*2^n+1[/B] (Pierpont [B]3^39*2^n+1[/B]) with [B]n=[1..10.000.000][/B].
The range was sieved up to p=114e12.
I use Sr1sieve, PFGW, LLR.

Now it is checked n=[1..1.000.000] (49.127 of 491.804 tests).

34 small primes found:

[CODE]4052555153018976267*2^2+1 is prime! (20 decimal digits) Time : 1.235 ms.
4052555153018976267*2^11+1 is prime! (22 decimal digits) Time : 2.454 ms.
4052555153018976267*2^82+1 is prime! (44 decimal digits) Time : 16.804 ms.
4052555153018976267*2^100+1 is prime! (49 decimal digits) Time : 28.723 ms.
4052555153018976267*2^106+1 is prime! (51 decimal digits) Time : 29.094 ms.
4052555153018976267*2^131+1 is prime! (59 decimal digits) Time : 48.247 ms.
4052555153018976267*2^160+1 is prime! (67 decimal digits) Time : 69.903 ms.
4052555153018976267*2^310+1 is prime! (112 decimal digits) Time : 503.097 ms.
4052555153018976267*2^490+1 is prime! (167 decimal digits) Time : 2.631 sec.
4052555153018976267*2^1052+1 is prime! (336 decimal digits) Time : 118.200 ms.
4052555153018976267*2^1486+1 is prime! (466 decimal digits) Time : 44.392 ms.
4052555153018976267*2^1532+1 is prime! (480 decimal digits) Time : 48.573 ms.
4052555153018976267*2^2995+1 is prime! (921 decimal digits) Time : 87.253 ms.
4052555153018976267*2^4295+1 is prime! (1312 decimal digits) Time : 67.959 ms.
4052555153018976267*2^8542+1 is prime! (2591 decimal digits) Time : 142.527 ms.
4052555153018976267*2^10531+1 is prime! (3189 decimal digits) Time : 183.830 ms.
4052555153018976267*2^11522+1 is prime! (3488 decimal digits) Time : 186.658 ms.
4052555153018976267*2^19975+1 is prime! (6032 decimal digits) Time : 465.413 ms.
4052555153018976267*2^23587+1 is prime! (7120 decimal digits) Time : 689.028 ms.
4052555153018976267*2^25202+1 is prime! (7606 decimal digits) Time : 751.311 ms.
4052555153018976267*2^26938+1 is prime! (8128 decimal digits) Time : 779.696 ms.
4052555153018976267*2^28448+1 is prime! (8583 decimal digits) Time : 929.404 ms.
4052555153018976267*2^30980+1 is prime! (9345 decimal digits) Time : 1.017 sec.
4052555153018976267*2^48527+1 is prime! (14627 decimal digits) Time : 2.692 sec.
4052555153018976267*2^64607+1 is prime! (19468 decimal digits) Time : 4.952 sec.
4052555153018976267*2^251027+1 is prime! (75586 decimal digits) Time : 81.627 sec.
4052555153018976267*2^283288+1 is prime! (85297 decimal digits) Time : 114.207 sec.
4052555153018976267*2^313411+1 is prime! (94365 decimal digits) Time : 126.248 sec.
4052555153018976267*2^374870+1 is prime! (112866 decimal digits) Time : 199.486 sec.
4052555153018976267*2^446486+1 is prime! (134425 decimal digits) Time : 303.894 sec.
4052555153018976267*2^513392+1 is prime! (154565 decimal digits) Time : 386.964 sec.
4052555153018976267*2^648871+1 is prime! (195349 decimal digits) Time : 570.565 sec.
4052555153018976267*2^759071+1 is prime! (228522 decimal digits) Time : 844.726 sec.
4052555153018976267*2^903875+1 is prime! (272113 decimal digits) Time : 1218.959 sec.[/CODE][/QUOTE]

The first two aren't proth primes, just fyi.

 Alex 2021-10-15 10:16

[QUOTE=dannyridel;590552]The first two aren't proth primes, just fyi.[/QUOTE]

Sure. So there is no word "proth" in the first post.

 dannyridel 2021-10-16 04:15

Yep. Any new primes?

 Alex 2021-10-31 13:32

[QUOTE=dannyridel;590717]Yep. Any new primes?[/QUOTE]

I have 8 active projects now (AP, xGF, Mersenne GPU TF, Near-repdigit, Proth, Pierpont, SG)
and report news in suitable thread, when pass a milestone or discover a new prime.
So see you soon :smile:

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