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-   -   Carol / Kynea Primes (https://www.mersenneforum.org/showthread.php?t=21251)

lalera 2016-05-17 18:52

hi,
here are the results for carol / kynea
b=136, 138, 140, 142, 146, 148, 150, 152
n=1 to 10000
[CODE]
--
(136^1-1)^2-2
(136^6-1)^2-2
(136^12+1)^2-2
(136^47+1)^2-2
--
(138^1+1)^2-2
(138^2+1)^2-2
(138^6-1)^2-2
(138^10-1)^2-2
(138^69+1)^2-2
(138^105-1)^2-2
(138^1716+1)^2-2
(138^4534-1)^2-2
(138^5407+1)^2-2
(138^6300+1)^2-2
(138^9489-1)^2-2
--
(140^1-1)^2-2
(140^4-1)^2-2
(140^5-1)^2-2
(140^29-1)^2-2
(140^41+1)^2-2
(140^155-1)^2-2
(140^382-1)^2-2
(140^395+1)^2-2
(140^485-1)^2-2
(140^1375-1)^2-2
(140^1528-1)^2-2
(140^4267+1)^2-2
(140^4456+1)^2-2
(140^5120+1)^2-2
(140^5396+1)^2-2
(140^5757-1)^2-2
--
(142^3+1)^2-2
(142^15-1)^2-2
(142^48+1)^2-2
(142^4869+1)^2-2
--
(146^1-1)^2-2
(146^3+1)^2-2
(146^20+1)^2-2
(146^35-1)^2-2
(146^37-1)^2-2
(146^403+1)^2-2
(146^2475+1)^2-2
(146^6965-1)^2-2
--
(148^2-1)^2-2
(148^4+1)^2-2
(148^20+1)^2-2
(148^30+1)^2-2
(148^43+1)^2-2
(148^60-1)^2-2
(148^112+1)^2-2
(148^255-1)^2-2
(148^422-1)^2-2
(148^528-1)^2-2
(148^1300-1)^2-2
--
(150^4+1)^2-2
(150^8-1)^2-2
(150^20-1)^2-2
(150^30-1)^2-2
(150^34+1)^2-2
(150^260-1)^2-2
--
(152^2+1)^2-2
(152^3-1)^2-2
(152^5-1)^2-2
(152^51+1)^2-2
(152^156+1)^2-2
(152^444+1)^2-2
(152^1263-1)^2-2
(152^1317-1)^2-2
--
[/CODE]

lalera 2016-05-17 19:40

hi,
here are the results for carol / kynea
b=154, 156, 158, 160, 162, 164, 166, 168
n=1 to 10000
[CODE]
--
(154^1+1)^2-2
(154^24-1)^2-2
(154^49+1)^2-2
(154^68-1)^2-2
(154^75+1)^2-2
(154^82+1)^2-2
(154^371+1)^2-2
(154^612+1)^2-2
(154^1495-1)^2-2
--
(156^1-1)^2-2
(156^136-1)^2-2
(156^152+1)^2-2
(156^461+1)^2-2
(156^1663-1)^2-2
--
(158^14-1)^2-2
(158^1893+1)^2-2
(158^8227+1)^2-2
--
(160^1+1)^2-2
(160^4-1)^2-2
(160^5+1)^2-2
(160^11-1)^2-2
(160^24+1)^2-2
(160^30+1)^2-2
(160^85-1)^2-2
(160^104+1)^2-2
(160^127-1)^2-2
(160^135+1)^2-2
(160^148+1)^2-2
(160^1104-1)^2-2
--
(162^1-1)^2-2
(162^12+1)^2-2
(162^82-1)^2-2
(162^386-1)^2-2
(162^447+1)^2-2
(162^3198-1)^2-2
(162^8342+1)^2-2
--
(164^2-1)^2-2
(164^6+1)^2-2
(164^15+1)^2-2
(164^1358-1)^2-2
(164^4967+1)^2-2
--
(166^2+1)^2-2
(166^3+1)^2-2
(166^78+1)^2-2
(166^321-1)^2-2
(166^9492-1)^2-2
--
(168^1+1)^2-2
(168^11-1)^2-2
(168^16+1)^2-2
(168^24-1)^2-2
(168^44+1)^2-2
(168^230+1)^2-2
(168^380-1)^2-2
(168^1140-1)^2-2
(168^6988+1)^2-2
--
[/CODE]

lalera 2016-05-17 20:28

hi,
here are the results for carol / kynea
b=170, 172, 174, 176, 178, 180, 182, 184
n=1 to 10000
[CODE]
--
(170^1-1)^2-2
(170^11+1)^2-2
(170^18-1)^2-2
(170^19-1)^2-2
(170^53+1)^2-2
(170^135-1)^2-2
(170^460-1)^2-2
(170^692+1)^2-2
(170^1059-1)^2-2
(170^1528-1)^2-2
(170^1653-1)^2-2
(170^2921+1)^2-2
(170^5141+1)^2-2
(170^9373-1)^2-2
--
(172^1+1)^2-2
(172^3+1)^2-2
(172^24+1)^2-2
(172^126+1)^2-2
(172^174-1)^2-2
(172^246-1)^2-2
(172^302-1)^2-2
(172^492+1)^2-2
(172^745+1)^2-2
(172^778+1)^2-2
(172^1191-1)^2-2
(172^1302-1)^2-2
(172^3680-1)^2-2
--
(174^1-1)^2-2
(174^2+1)^2-2
(174^10+1)^2-2
(174^15-1)^2-2
(174^20-1)^2-2
(174^96-1)^2-2
(174^354-1)^2-2
(174^396+1)^2-2
(174^1100+1)^2-2
(174^3894+1)^2-2
--
(176^1+1)^2-2
(176^6-1)^2-2
(176^7+1)^2-2
(176^29-1)^2-2
(176^2001-1)^2-2
(176^3385+1)^2-2
(176^7360-1)^2-2
(176^7566+1)^2-2
(176^8866+1)^2-2
--
(178^1-1)^2-2
(178^3+1)^2-2
(178^4+1)^2-2
(178^6+1)^2-2
(178^9-1)^2-2
(178^174-1)^2-2
(178^313-1)^2-2
(178^609-1)^2-2
(178^839+1)^2-2
--
(180^14+1)^2-2
(180^42-1)^2-2
(180^189+1)^2-2
(180^218+1)^2-2
(180^251-1)^2-2
(180^360+1)^2-2
--
(182^1+1)^2-2
(182^5+1)^2-2
(182^6+1)^2-2
(182^227+1)^2-2
(182^310-1)^2-2
(182^1258-1)^2-2
(182^1348+1)^2-2
--
(184^1-1)^2-2
(184^2+1)^2-2
(184^12+1)^2-2
(184^51+1)^2-2
(184^160-1)^2-2
(184^671+1)^2-2
(184^2907-1)^2-2
(184^3417-1)^2-2
--
[/CODE]

lalera 2016-05-17 21:15

hi,
here are the results for carol / kynea
b=186, 188, 190, 192, 194, 198, 200
n=1 to 10000
[CODE]
--
(186^2-1)^2-2
(186^6+1)^2-2
(186^12-1)^2-2
(186^206-1)^2-2
(186^293-1)^2-2
(186^459+1)^2-2
(186^811+1)^2-2
(186^1968+1)^2-2
--
(188^2+1)^2-2
(188^5+1)^2-2
(188^27-1)^2-2
(188^39-1)^2-2
(188^55-1)^2-2
(188^77-1)^2-2
(188^926+1)^2-2
(188^1406-1)^2-2
(188^2225+1)^2-2
(188^2544-1)^2-2
(188^3863+1)^2-2
(188^5495+1)^2-2
(188^6052-1)^2-2
(188^6292-1)^2-2
--
(190^2-1)^2-2
(190^1+1)^2-2
(190^4+1)^2-2
(190^6-1)^2-2
(190^18-1)^2-2
(190^40-1)^2-2
(190^71+1)^2-2
(190^262+1)^2-2
(190^431-1)^2-2
(190^2841+1)^2-2
--
(192^1-1)^2-2
(192^2-1)^2-2
(192^7-1)^2-2
(192^26-1)^2-2
(192^39-1)^2-2
(192^1017+1)^2-2
(192^7989+1)^2-2
--
(194^3+1)^2-2
(194^29-1)^2-2
(194^180-1)^2-2
(194^5007-1)^2-2
--
(198^6+1)^2-2
(198^12+1)^2-2
(198^103-1)^2-2
(198^118-1)^2-2
--
(200^3+1)^2-2
(200^20-1)^2-2
(200^33+1)^2-2
(200^36-1)^2-2
(200^37+1)^2-2
(200^5448+1)^2-2
--
[/CODE]

lalera 2016-05-17 22:22

hi,
here are the results for carol / kynea
b=80, 82, 84, 86, 88, 90
n=1000 to 30000
--
(80^1445+1)^2-2
(80^22631+1)^2-2
--
(82^1074-1)^2-2
(82^1212-1)^2-2
(82^1866+1)^2-2
(82^20148-1)^2-2
--
(84^1253-1)^2-2
(84^1922-1)^2-2
(84^2613-1)^2-2
(84^4162+1)^2-2
(84^5582-1)^2-2
(84^14493+1)^2-2
--
(86^1120-1)^2-2
(86^2053-1)^2-2
(86^11270+1)^2-2
--
(88^1072+1)^2-2
(88^5100+1)^2-2
(88^28032+1)^2-2
--
(90^1105-1)^2-2
(90^2186-1)^2-2
(90^3120+1)^2-2
(90^6957-1)^2-2
--

rogue 2016-05-18 12:43

lalera, thanks for your work on this. I have made all of the updates. Please double-check to verify that I didn't make any mistakes.

It is curious that all bases for both Carol and Kynea have at least one prime. I was expecting one or more to have no primes for n < 1000.

Batalov 2016-05-18 17:25

There are [B]b=640 and 688[/B] without "easy" primes (and then a bit above them, [B]b=1656[/B], [STRIKE]1852, 1950[/STRIKE]). The rest of b<=2000 have small primes.
Some of the larger "but still small" first primes:
[CODE](926^699-1)^2-2
(368^969+1)^2-2
(970^1008-1)^2-2
(1318^1013+1)^2-2
(982^1053-1)^2-2
(1432^1578-1)^2-2
(1038^2107-1)^2-2
(1388^7458-1)^2-2
(1452^574+1)^2-2
(1466^4249-1)^2-2
(1468^4351+1)^2-2
(1484^581-1)^2-2
(1614^2907+1)^2-2
(1852^6341-1)^2-2
[URL="http://factordb.com/index.php?id=1100000000838999788"](1950^8442+1)^2-2[/URL]
(1992^472+1)^2-2[/CODE]

rogue 2016-05-18 18:04

[QUOTE=Batalov;434317]There are [B]b=640 and 688[/B] without "easy" primes (and then a bit above them, b=1388 and 1432). The rest of b<=1000 have small primes.
Some of the larger "but still small" first primes:
[CODE](926^699-1)^2-2
(368^969+1)^2-2
(970^1008-1)^2-2
(1318^1013+1)^2-2
(982^1053-1)^2-2
(1038^2107-1)^2-2[/CODE][/QUOTE]

Did you actually search these bases or did you find a list elsewhere?

Batalov 2016-05-18 18:14

I'd quickly searched them just now. A list generated with GP was
[CODE]? forstep(b=2,2000,2,if(ispower(b)>1,next);q=0;for(n=1,400,if(ispseudoprime((b^n-1)^2-2),q=1;break);if(ispseudoprime((b^n+1)^2-2),q=1;break));if(!q,print1(" "b)))
368 640 688 926 970 982 1038 1270 1318 1388 1432 1452 1466 1468 1484 1614 1656 1852 1950 1992[/CODE]
and then I sieved the smaller ones (up to 1432) to n<=10000 and started pfgw. After a few minutes only b=640, 688, 1388 survived.

I guess, I will reserve b=640, 688 to n <= 30000 for starters, now.

rogue 2016-05-18 21:06

Did you generate a list of primes for the bases? If so, I will likely need to offload the list elsewhere. Anyone know of a good place that I can host some web pages for free and not have to deal with ads?

Batalov 2016-05-18 21:25

No lists, because the GP scripts simply quits after very small primes (and that saves all the time; in particular, many bases have very small primes: n=1, 2, 3).
I will generate a proper list (most likely, just one (k,c) pair, of course) for b=640, 688.


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