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-   And now for something completely different (https://www.mersenneforum.org/forumdisplay.php?f=119)
-   -   Carol / Kynea Primes (https://www.mersenneforum.org/showthread.php?t=21251)

 rogue 2016-04-15 17:28

Carol / Kynea Primes

Please go [URL="http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm"]here[/URL] to get a complete list of Carol and Kynea primes. You can continue to post new primes in this thread.

[url]https://en.wikipedia.org/wiki/Carol_number[/url]
[url]https://en.wikipedia.org/wiki/Kynea_number[/url]

 NorbSchneider 2016-04-16 22:26

Carol / Kynea Primes

I search base 38 up to n=70,000.

I reached n=5,000 and found 7 new PRPs:
(Carol)
(38^1-1)^2-2, 4 digits
(38^2-1)^2-2, 7 digits
(38^13-1)^2-2, 42 digits
(38^560-1)^2-2, 1770 digits

(Kynea)
(38^6+1)^2-2, 19 digits
(38^279+1)^2-2, 882 digits
(38^3490+1)^2-2, 11027 digits

 wombatman 2016-04-18 18:39

Base 26

Double-checked the range from n=1 to n=14083. Found the following primes (3-PRP and N-1/N+1 via PFGW):

[CODE](26^8+1)^2-2
(26^78+1)^2-2
(26^79+1)^2-2
(26^111+1)^2-2
(26^159-1)^2-2
(26^879-1)^2-2
(26^4744-1)^2-2
(26^5276+1)^2-2
(26^5602-1)^2-2
(26^8226+1)^2-2
[/CODE]

 lalera 2016-04-18 22:28

[QUOTE=wombatman;431884]Double-checked the range from n=1 to n=14083. Found the following primes (3-PRP and N-1/N+1 via PFGW):

[CODE](26^8+1)^2-2
(26^78+1)^2-2
(26^79+1)^2-2
(26^111+1)^2-2
(26^159-1)^2-2
(26^879-1)^2-2
(26^4744-1)^2-2
(26^5276+1)^2-2
(26^5602-1)^2-2
(26^8226+1)^2-2
[/CODE][/QUOTE]

... and
(26^1+1)^2-2
(26^2+1)^2-2

 axn 2016-04-20 08:30

Update on bases 12 and 18

base 12 searched upto 30,000 (continuing to 50,000)
[CODE](12^1+1)^2-2
(12^2+1)^2-2
(12^3-1)^2-2
(12^8+1)^2-2
(12^29-1)^2-2
(12^51-1)^2-2
(12^60+1)^2-2
(12^513+1)^2-2
(12^1047+1)^2-2
(12^7021+1)^2-2
(12^7506+1)^2-2
(12^7824-1)^2-2
(12^15456-1)^2-2
(12^22614-1)^2-2
(12^28312-1)^2-2[/CODE]

base 18 searched upto 26,000 (sieved upto 100,000, continuing till i'm bored)
[CODE](18^1+1)^2-2
(18^2-1)^2-2
(18^8-1)^2-2
(18^10+1)^2-2
(18^21+1)^2-2
(18^25+1)^2-2
(18^30-1)^2-2
(18^31+1)^2-2
(18^98-1)^2-2
(18^110-1)^2-2
(18^185-1)^2-2
(18^912-1)^2-2
(18^1083+1)^2-2
(18^2514-1)^2-2
(18^4074-1)^2-2
(18^10208-1)^2-2
(18^15123-1)^2-2
(18^19395-1)^2-2
[/CODE]
All proven primes, and available in factordb (the larger ones apparently cannot be proven in factordb, even though N+1 is adequately factored).

 axn 2016-04-20 08:55

[QUOTE=rogue;431652]
Carol Primes (b^n-1)^2-2:
[code]
base
2 (n < 100000) 2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637
2 (n > 100000) 100224, 108127, 110953, 175749, 185580, 226749, 248949, 253987, 520363
6 6, 7, 20, 47, 255, 274, 279, 308, 1162, 2128, 3791, 9028, 9629, 10029, 13202, 38660, 46631, 48257
10 8, 21, 123, 4299, 6128, 11760, 18884, 40293
14 1, 6, 13, 45, 74, 240, 553, 12348, 13659
22 8, 35, 88, 503, 8642, 8743, 14475
26 159, 879, 4744, 5602
38 1, 2, 13, 560
204 5, 7, 40, 11867, 14458, 1752, 18929
[/code]

Kynea Primes (b^n+1)^2-2:
[code]
base
2 (n < 100000) 1, 2, 3, 5, 8, 9, 12, 15, 17,18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, 491, 501, 507, 555, 591, 680, 800, 1070, 1650, 2813, 3281, 4217, 5153, 6287, 6365, 10088, 10367, 37035, 45873, 69312
2 (n > 100000) 102435, 106380, 108888, 110615, 281621, 369581, 376050, 442052
6 9, ,12, 30, 49, 56, 115, 118, 376, 432, 1045, 1310, 6529, 7768, 8430, 21942, 26930, 33568
10 22, 123, 351, 1061
14 1, 5, 60, 72, 118, 181, 245, 310, 498, 820, 962, 2212, 3928, 584, 5937
22 3, 166, 814, 1851, 2197, 3172, 3865, 19791
26 1, 2, 8, 78, 79, 111, 5276, 8226
38 6, 279, 3490
204 40, 3645, 23750
[/code][/QUOTE]
Mark, some of the smaller primes seem to be missing from this list (possibly because cksieve sieves them out). I have generated all primes for 6 <= b <= 100, n <= 64 using PARI/GP.

Carol
[CODE]6:1,2,6,7,20,47
10:1,8,21
12:3,29,51
14:1,6,13,45
18:2,8,30
20:1,2,53
22:1,8,35
24:2,27
26:
28:1,22
30:1,6,19,30
34:1,4
38:1,2,13
40:4,15,39
42:3,6,14,15,29
44:1,7,30
46:12
48:1,2,4,6,12,13
50:1,3,4,9,31
52:2,14,24
54:9,17
56:1,2,3,11
58:
60:2,5
62:1
66:12
68:4,59
70:1,5,9,18,24
72:1,9
74:
76:1,2,32,37,51
78:1,2
80:
82:24
84:4
86:3,39
88:2,5,9
90:1,5,43
92:3,4,6,60
94:1,2
96:5
98:2,21[/CODE]

Kynea
[CODE]6:1,2,3,4,9,12,30,49,56
10:22
12:1,2,8,60
14:1,5,60
18:1,10,21,25,31
20:1,15,44
22:3
24:24
26:1,2,8
28:1,2,11,15
30:2,3,57
34:1,2,14,29,61
38:6
40:2,49
42:1,3,4
44:3
46:1,54
48:1
50:4,38
52:3,5
54:1
56:8,14
58:2,21,35
60:1,9,21,49
62:1,2
66:
68:1,3,6,32
70:1,7,11
72:
74:1,3
76:1,2,3,22,29
78:3,5,13
80:29,34,45
82:3,9
84:
86:8,18,36
88:1,40
90:11
92:1,2
94:5
96:2,13,15,54
98:4,12,35[/CODE]

 Batalov 2016-04-20 17:20

Here is my small extension to Anand's table above. So far I have these (I run them all together sorted by size)
Carol Primes (b^n-1)^2-2:
[CODE]
10: 1,8,21,123,4299,6128,11760,18884,40293 (known) [79631]
20: 1,2,53,183,1281,1300,8041,29936 [61385]
30: 1,6,19,30,166,495,769,826,1648,3993 [53812]
40: 4,15,39,138,2153,4084,5639 [50000]
[/CODE]Kynea Primes (b^n+1)^2-2:
[CODE]
10: 22,351,1061 (known) [79631]
20: 1,15,44,77,141,208,304,1169,3359,5050,22431,34935 [61385]
30: 2,3,57,129,171,9837,30359 [53812]
40: 2,49,144,825,2856,2996,5166,7824,9392,40778 [50000]
[/CODE]

 lalera 2016-04-21 17:34

hi,
here are the results for carol / kynea
b=6, n=1 to 50000
[CODE]
(6^1-1)^2-2
(6^2-1)^2-2
(6^6-1)^2-2
(6^1+1)^2-2
(6^2+1)^2-2
(6^3+1)^2-2
(6^4+1)^2-2
(6^7-1)^2-2
(6^9+1)^2-2
(6^12+1)^2-2
(6^20-1)^2-2
(6^30+1)^2-2
(6^47-1)^2-2
(6^49+1)^2-2
(6^56+1)^2-2
(6^115+1)^2-2
(6^118+1)^2-2
(6^255-1)^2-2
(6^274-1)^2-2
(6^279-1)^2-2
(6^308-1)^2-2
(6^376+1)^2-2
(6^432+1)^2-2
(6^1045+1)^2-2
(6^1162-1)^2-2
(6^1310+1)^2-2
(6^2128-1)^2-2
(6^3791-1)^2-2
(6^6529+1)^2-2
(6^7768+1)^2-2
(6^8430+1)^2-2
(6^9028-1)^2-2
(6^9629-1)^2-2
(6^10029-1)^2-2
(6^13202-1)^2-2
(6^21942+1)^2-2
(6^26930+1)^2-2
(6^33568+1)^2-2
(6^38660-1)^2-2
(6^46631-1)^2-2
(6^48257-1)^2-2
[/CODE]
please keep my reservation for base 6

 lalera 2016-04-21 18:08

hi,
here are the results for carol / kynea
b=74, n=1 to 20000
[CODE]
(74^1+1)^2-2
(74^3+1)^2-2
(74^183-1)^2-2
(74^694+1)^2-2
(74^2300-1)^2-2
(74^3855-1)^2-2
[/CODE]
continuing

 henryzz 2016-04-28 22:36

Base 24 tested until n=10k
primes:
[CODE](24^24+1)^2-2
(24^27-1)^2-2
(24^92-1)^2-2
(24^321+1)^2-2
(24^971+1)^2-2
(24^984+1)^2-2
(24^4950-1)^2-2[/CODE]

 WMHalsdorf 2016-04-29 00:10

(120^6-1)^2-2
(120^7-1)^2-2
(120^11-1)^2-2
(120^12-1)^2-2
(120^12+1)^2-2
(120^16+1)^2-2
(120^168-1)^2-2
(120^236+1)^2-2
(120^1501+1)^2-2
(120^2249+1)^2-2
(120^2750-1)^2-2
(120^3421+1)^2-2

Currently up to 7000

 lalera 2016-04-29 00:19

hi,
here are the results for carol / kynea
b=74, n=20000 to 40000
(74^39227-1)^2-2
continuing

 Batalov 2016-05-05 06:23

After a bit of a side trip with xy-yx's, continued with the CK's search and found a slightly larger one: [URL="http://factordb.com/index.php?id=1100000000838084937"](20^72820-1)^2-2[/URL]
(current limits are 10: [95000], 20: [74000], 30: [65000], 40: [60000])

 lalera 2016-05-05 12:35

hi,
here are the results for carol / kynea
b=6, n=50000 to 100000
(6^50800+1)^2-2
continuing

 lalera 2016-05-05 14:06

hi,
here are the results for
b=70, n=1000 to 10000
no prime
b=72, n=1000 to 10000
(72^1920+1)^2-2
(72^6831+1)^2-2

 wombatman 2016-05-05 15:53

For b = 26:

Tested through n=14,084 - 34,000 (would be higher, but I oversieved a bit)

Primes:
[CODE](26^19545+1)^2-2[/CODE]

Continuing on to n=100,000

 rogue 2016-05-05 16:17

[QUOTE=wombatman;433150]For b = 26:

Tested through n=14,084 - 34,000 (would be higher, but I oversieved a bit)

Primes:
[CODE](26^19545+1)^2-2[/CODE]

Continuing on to n=100,000[/QUOTE]

Thanks for the update. What did you sieve to? How many tests do you have to do in that range?

 wombatman 2016-05-05 16:31

Sieve is to P=~3.22e12. I stopped when I saw that the seconds per factor were over 500, and I figured that the PRP tests would be faster than that.

Assuming the range you're referring to is from n=34,000 to n=100,000, there are approximately 11,000 tests or so to go.

 rogue 2016-05-05 17:04

[QUOTE=wombatman;433154]Sieve is to P=~3.22e12. I stopped when I saw that the seconds per factor were over 500, and I figured that the PRP tests would be faster than that.

Assuming the range you're referring to is from n=34,000 to n=100,000, there are approximately 11,000 tests or so to go.[/QUOTE]

Then you might have not sieved deeply enough. You probably want the removal rate to be between 1000 and 1500 seconds per factor, depending upon the hardware you are running. Also, are you using the latest cksieve? It is about 50% faster than the earlier ones.

 wombatman 2016-05-05 20:44

I am, yes. Current PRP tests are taking ~250 seconds per test, compared to finding factors at 500+ seconds per factor. I know that at some point it will make sense to sieve more, but I was thinking that would occur when the PRP test was in the neighborhood of the sec/factor time.

Would you recommend sieving more now?

 rogue 2016-05-05 20:57

[QUOTE=wombatman;433174]I am, yes. Current PRP tests are taking ~250 seconds per test, compared to finding factors at 500+ seconds per factor. I know that at some point it will make sense to sieve more, but I was thinking that would occur when the PRP test was in the neighborhood of the sec/factor time.

Would you recommend sieving more now?[/QUOTE]

No. You can. See what the speed is around n=50,000. You might consider doing more sieving when you get there.

 wombatman 2016-05-05 21:20

Will do. Thanks.

 lalera 2016-05-07 20:23

hi,
i found a mistake in the table of carol primes
b=96 n=5 is prime and not n=6
and may i remind you that i did already the range b=72, n=1000 to 10000 (post #15)

 rogue 2016-05-07 21:49

[QUOTE=lalera;433306]hi,
i found a mistake in the table of carol primes
b=96 n=5 is prime and not n=6
and may i remind you that i did already the range b=72, n=1000 to 10000 (post #15)[/QUOTE]

I fixed the on for base 96, a typo on my part. Completed ranges are marked in the other thread.

 lalera 2016-05-07 22:26

[QUOTE=rogue;433310]I fixed the on for base 96, a typo on my part. Completed ranges are marked in the other thread.[/QUOTE]

hi,
i do mean that the range b=72, n=1000 to 10000 (post #15) with the two primes
is not updated in the primes list and also not updated in the reservation table in the other thread
the two primes are for base 72 and not for base 70 - the kynea table is wrong

 rogue 2016-05-07 23:59

I see. Fixed.

 axn 2016-05-11 05:27

[URL="http://factordb.com/index.php?id=1100000000838567960"](12^47014-1)^2-2[/URL]

Thanks Serge for the tip!

 NorbSchneider 2016-05-16 20:51

Base 38 searched to n = 30,000 1 new PRP -> (38^28933-1)^2-2,
continuing to n=70,000.

 WMHalsdorf 2016-05-16 23:13

(120^7112+1)^2-2
(120^8260+1)^2-2

Checked to 10k

 lalera 2016-05-17 14:13

hi,
here are the results for carol / kynea
b=74, n=40000 to 69000
(74^56039-1)^2-2
continuing

 lalera 2016-05-17 14:32

hi,
here are the results for carol / kynea
b=92, n=1000 to 30000
(92^1795+1)^2-2
(92^2831-1)^2-2
(92^4281-1)^2-2
b=94, n=1000 to 30000
(94^3305+1)^2-2
(94^11045+1)^2-2
(94^13278-1)^2-2
(94^16021-1)^2-2
(94^20654-1)^2-2
b=96, n=1000 to 30000
no prime
b=98, n=1000 to 30000
(98^2588-1)^2-2
(98^5876-1)^2-2
(98^28725+1)^2-2

 lalera 2016-05-17 16:36

hi,
here are the results for carol / kynea
b=102, 104, 106, 108, 110, 112, 114, 116
n=1 to 10000
--
(102^2-1)^2-2
(102^1+1)^2-2
(102^3+1)^2-2
(102^27+1)^2-2
(102^33+1)^2-2
(102^92-1)^2-2
(102^165-1)^2-2
(102^264+1)^2-2
(102^292+1)^2-2
(102^492-1)^2-2
(102^1630+1)^2-2
(102^1993+1)^2-2
(102^2628-1)^2-2
(102^2879-1)^2-2
(102^4676-1)^2-2
(102^5603-1)^2-2
--
(104^1-1)^2-2
(104^6+1)^2-2
(104^53-1)^2-2
(104^59-1)^2-2
(104^85+1)^2-2
(104^87+1)^2-2
(104^638+1)^2-2
(104^832+1)^2-2
(104^2772-1)^2-2
(104^2798+1)^2-2
(104^3867-1)^2-2
(104^6328-1)^2-2
(104^6365+1)^2-2
--
(106^2-1)^2-2
(106^1+1)^2-2
(106^3-1)^2-2
(106^6+1)^2-2
(106^25+1)^2-2
(106^27+1)^2-2
(106^46-1)^2-2
(106^47-1)^2-2
(106^104-1)^2-2
(106^219+1)^2-2
(106^539+1)^2-2
(106^1641+1)^2-2
(106^2046+1)^2-2
(106^2922-1)^2-2
--
(108^1-1)^2-2
(108^3+1)^2-2
(108^5+1)^2-2
(108^24+1)^2-2
(108^1078+1)^2-2
(108^2104+1)^2-2
(108^3807-1)^2-2
(108^8292+1)^2-2
--
(110^57+1)^2-2
(110^705-1)^2-2
(110^1245+1)^2-2
(110^1870-1)^2-2
(110^2852+1)^2-2
(110^6403+1)^2-2
--
(112^2-1)^2-2
(112^6-1)^2-2
(112^24+1)^2-2
(112^527+1)^2-2
(112^542-1)^2-2
(112^1656-1)^2-2
(112^1813+1)^2-2
--
(114^3-1)^2-2
(114^72-1)^2-2
(114^156-1)^2-2
(114^368+1)^2-2
(114^406-1)^2-2
(114^472-1)^2-2
(114^3270-1)^2-2
(114^6873-1)^2-2
--
(116^1+1)^2-2
(116^12+1)^2-2
(116^29-1)^2-2
(116^38-1)^2-2
(116^1272+1)^2-2
(116^1852+1)^2-2
--

 lalera 2016-05-17 17:47

hi,
here are the results for carol / kynea
b=118, 120, 122, 124, 126, 130, 132, 134
n=1 to 10000
[CODE]
--
(118^1-1)^2-2
(118^1+1)^2-2
(118^4-1)^2-2
(118^17+1)^2-2
(118^40+1)^2-2
(118^55-1)^2-2
(118^150-1)^2-2
(118^223-1)^2-2
(118^886-1)^2-2
(118^3381-1)^2-2
--
(120^1-1)^2-2
(120^1+1)^2-2
(120^6-1)^2-2
(120^7-1)^2-2
(120^11-1)^2-2
(120^12+1)^2-2
(120^12-1)^2-2
(120^16+1)^2-2
(120^168-1)^2-2
(120^236+1)^2-2
(120^1501+1)^2-2
(120^2249+1)^2-2
(120^2750-1)^2-2
(120^3421+1)^2-2
(120^7112+1)^2-2
(120^8260+1)^2-2
--
(122^1-1)^2-2
(122^2-1)^2-2
(122^11+1)^2-2
(122^45+1)^2-2
(122^244+1)^2-2
(122^489-1)^2-2
(122^499-1)^2-2
(122^1474+1)^2-2
(122^2552-1)^2-2
(122^2565-1)^2-2
--
(124^19+1)^2-2
(124^36+1)^2-2
(124^728+1)^2-2
(124^4836-1)^2-2
--
(126^1+1)^2-2
(126^2+1)^2-2
(126^20-1)^2-2
(126^37+1)^2-2
(126^58+1)^2-2
(126^84+1)^2-2
(126^93-1)^2-2
(126^126+1)^2-2
(126^276+1)^2-2
(126^2318+1)^2-2
(126^2877+1)^2-2
(126^4514+1)^2-2
(126^5924-1)^2-2
(126^6212+1)^2-2
--
(130^1+1)^2-2
(130^135-1)^2-2
(130^178+1)^2-2
(130^672+1)^2-2
--
(132^1-1)^2-2
(132^2-1)^2-2
(132^10-1)^2-2
(132^13+1)^2-2
(132^57-1)^2-2
(132^236-1)^2-2
(132^665-1)^2-2
(132^1876-1)^2-2
--
(134^1+1)^2-2
(134^4-1)^2-2
(134^11+1)^2-2
(134^16-1)^2-2
(134^42+1)^2-2
(134^65-1)^2-2
(134^81-1)^2-2
(134^93-1)^2-2
(134^161-1)^2-2
(134^204+1)^2-2
(134^1171+1)^2-2
(134^1932-1)^2-2
(134^2677-1)^2-2
(134^4681-1)^2-2
(134^4695-1)^2-2
--
[/CODE]

 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.

 Batalov 2016-05-19 01:58

[QUOTE=rogue;434332]Anyone know of a good place that I can host some web pages for free and not have to deal with ads?[/QUOTE]
You could make an additional page (or pages) at prothsearch.net, maybe?

 a1call 2016-05-19 03:57

[QUOTE=rogue;434332]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?[/QUOTE]

If interested, I can provide you with a subdomain from one of my domains.
I have 19.lc registered.
Choose a subdomain as short as you would like and I will set it up.

Notes:

* I can not guarantee a delivery time
* I can not guarantee duration of service

So if interested please let me know and make sure you have backups.

 rogue 2016-05-19 12:46

Thanks for your suggestions. MIke (Xyzzy) has offered me space here at mersenneforum, which is the best of all worlds. I had considered prothsearch, but I don't own the domain and with Ray being gone and Wilfred getting older, I don't know how much longer that domain will be around. Also, most of the work is being done by PrimeGrid, so I have been considering locking users from making reservations.

 lalera 2016-05-19 13:37

hi,
here are the results for carol / kynea
b=66, n=1000 to 30000
(66^1751+1)^2-2
(66^1902-1)^2-2
(66^9832+1)^2-2
(66^28522+1)^2-2
b=68, n=1000 to 30000
(68^2272-1)^2-2
(68^2634-1)^2-2
(68^2960+1)^2-2
(68^3014+1)^2-2
(68^16522-1)^2-2

 wombatman 2016-05-19 18:06

For b=26, no new primes through n=50k. Continuing on to n=100k.

 Batalov 2016-05-19 21:12

For b=688, [URL="http://factordb.com/index.php?id=1100000000839008688"](688^10160+1)^2-2[/URL] is prime!

For b<=2000, only b=640 and b=1656 remain without known primes.
I reserve b=1656 to n<=20000 and release b=688 at n=11000.

 paulunderwood 2016-05-21 18:28

top5000

Congrats Serge for finding the near square prime [URL="http://primes.utm.edu/primes/page.php?id=121680"](178^87525 - 1)^2 - 2[/URL] with 393937 decimal digits! :smile:

 rogue 2016-05-21 19:41

[QUOTE=paulunderwood;434582]Congrats Serge for finding the near square prime [URL="http://primes.utm.edu/primes/page.php?id=121680"](178^87525 - 1)^2 - 2[/URL] with 393937 decimal digits! :smile:[/QUOTE]

I was wondering. Chris Caldwell's page sent an automated e-mail for cksieve as a program and I wasn't the one to do it.

Thanks Serge and congrats.

 Batalov 2016-05-22 01:05

Sometime last week I've thought about attacking a recordable (Top5000) C-K prime.
I've sieved b=10, 20, 30, 40 far enough, but altogether they didn't have enough sub-1-hour candidates after sieve to warrant a success (After a certain size candidates start taking more than an hour each, and then quickly 1.5 then 2 hours.) So to bridge the path to certain success, I additionally sieved 102<=b<=202 (but not powers) to 1e11 -- for a small slice just above 389K decimal digits. Then for technical reasons I chose only 36 thickest bases and sieved them up to 2e12. Then I merged all files and sorted by size and started ~ a hundred PFGW instances. This morning a prime popped up for b=178.

I will test through the weekend, then stop, and will post all tested ranges explicitly; they can be added to control tables in square brackets as some Proth and Riesel tables on certain sites are recorded.

 Batalov 2016-05-23 00:40

And now, a Kynea prime for b=30, to make a nice pair.
(30[SUP]157950[/SUP] + 1)[SUP]2[/SUP] - 2 [URL="http://primes.utm.edu/primes/page.php?id=121686"]is prime[/URL]

 paulunderwood 2016-05-23 00:53

Congrats :banana:

 rogue 2016-05-23 01:12

[QUOTE=Batalov;434661]And now, a Kynea prime for b=30, to make a nice pair.
(30[SUP]157950[/SUP] + 1)[SUP]2[/SUP] - 2 [URL="http://primes.utm.edu/primes/page.php?id=121686"]is prime[/URL][/QUOTE]

:censored:, I mean :fusion:.

You're getting ahead of me with bases that don't require as many tests to find a Top 5000 prime.

I have not put together a webpage yet as I've been busy with outdoor stuff. The weather this past weekend was fantastic here. I hope to get something in place by Memorial Day (next week Monday for those non-Americans out there).

 lalera 2016-05-23 09:06

hi,
here are the results for carol / kynea
b=28, n=1000 to 30000
(28^2520-1)^2-2
(28^5048+1)^2-2
(28^6492-1)^2-2
(28^6577-1)^2-2
(28^22960-1)^2-2
(28^24990+1)^2-2
(28^25528-1)^2-2

 Batalov 2016-05-24 07:09

[QUOTE=Batalov;434409]For b<=2000, only [B]b=640 and b=1656[/B] remain without known primes.[/QUOTE]
For b<=3000, in addition to b=640 (up to n<=40K) and b=1656 (up to n<=35K), there are three more b = {2264, 2482, 2634} that remain without known C-K primes (up to n<=5100).

For b=2026, the smallest prime is (2026^5526+1)^2-2, the rest have a prime under n<=1000.

 rogue 2016-05-25 00:10

I'm starting to put together a website for this search. Take a look at [url]http://www.mersenneforum.org/rogue/ckps.html[/url] and give me some feedback.

 Batalov 2016-05-26 06:31

[B]Update:[/B]
For b<=3000, there are [STRIKE]four[/STRIKE] [COLOR=Blue]three[/COLOR] cases: b=1656 (up to n<=45K) and b = {[STRIKE]2264,[/STRIKE][SUP][COLOR=Blue]*[/COLOR][/SUP] 2482, 2634} that remain without known C-K primes (up to n<=8000).

However, for b=640, (640^44940+1)^2-2 is prime (252220 digits),
[B]so all b<1656 have at least one known C-K prime!
___________________
[/B][COLOR=Blue][SUP]*[/SUP] (2264^10098+1)^2-2 is prime[/COLOR]

 lalera 2016-05-27 08:45

hi,
here are the results for carol / kynea
b=72, n=10000 to 40000
(72^37930-1)^2-2

 lalera 2016-05-28 20:20

hi,
here are the results for carol / kynea
b=6, n=100000 to 120000
(6^117991-1)^2-2

 lalera 2016-05-29 14:56

hi,
here are the results for carol / kynea
b=70, n=10000 to 40000
(70^10744-1)^2-2
(70^11260-1)^2-2
(70^19370-1)^2-2

 rogue 2016-05-29 19:43

lalera, did you only find Carol primes in those bases?

 lalera 2016-05-30 09:00

[QUOTE=rogue;435098]lalera, did you only find Carol primes in those bases?[/QUOTE]

yes

 lalera 2016-05-30 09:29

hi,
here are the results for carol / kynea
b=76, n=1000 to 40000
(76^2306+1)^2-2
(76^3081-1)^2-2
(76^35617-1)^2-2
(76^36090-1)^2-2
b=78, n=1000 to 40000
(78^6429-1)^2-2
(78^6853+1)^2-2

 rogue 2016-05-30 23:19

(2^621443+1)^2-2 is prime! At 374176 digits, it is a about 14,000 digits short of the Top 5000.

 Batalov 2016-05-31 01:13

Arrgh! :chappy:
Still a very decent number!

 paulunderwood 2016-05-31 02:49

[QUOTE=rogue;435175](2^621443+1)^2-2 is prime! At 374176 digits, it is a about 14,000 digits short of the Top 5000.[/QUOTE]

Congrats. The next one will be in the top5000. :smile:

 Batalov 2016-06-01 01:51

(2482^18735+1)^2-2 is prime.

Now we only have two bases b=1656 (n<=70000) and b=2634 (n<=66000) that have no known primes under b<=3000.

____
[COLOR=Gray]EDIT: updated their search limits.[/COLOR]

 a1call 2016-06-03 14:03

Please go [URL="http://www.mersenneforum.org/rogue/ckps.html"]here[/URL] to get a complete list of Carol and Kynea primes. You can continue to post new primes in this thread.[/QUOTE]

I have taken the liberty of setting up the following short redirect, should anyone care to use it:

[url]http://ckps.19.lc[/url]

If there is an objection I will cancel the redirection.

Preferably by at least a PM.

 lalera 2016-06-06 15:40

hi,
here are the results for carol / kynea
b=60, n=1000 to 30000
(60^1717+1)^2-2
(60^3882+1)^2-2
(60^6665-1)^2-2
(60^11307+1)^2-2
(60^11549-1)^2-2
b=62, n=1000 to 30000
(62^1573-1)^2-2
(62^2183+1)^2-2
(62^3818+1)^2-2
(62^8800+1)^2-2
(62^11080-1)^2-2
(62^13352+1)^2-2
(62^17625+1)^2-2

 rogue 2016-06-14 21:17

[URL="http://primes.utm.edu/primes/page.php?id=121779"](2^653490 - 1)^2 - 2[/URL] is prime! It will be in the 3700 range after it is verified.

 paulunderwood 2016-06-14 21:40

[QUOTE=rogue;436236][URL="http://primes.utm.edu/primes/page.php?id=121779"](2^653490 - 1)^2 - 2[/URL] is prime! It will be in the 3700 range after it is verified.[/QUOTE]

Congrats on a top5000 near-square prime :banana:

 rogue 2016-06-18 13:05

(2^661478+1)^2-2 is prime at 398250 digits. :fusion:

 paulunderwood 2016-06-18 14:39

[QUOTE=rogue;436477](2^661478+1)^2-2 is prime at 398250 digits. :fusion:[/QUOTE]

Congrats again :smile:

 wombatman 2016-06-28 05:29

For base 26:

[CODE](26^74387-1)^2-2 is 3-PRP! (1279.8199s+0.0082s)[/CODE]

Still working to n=100,000.

 wombatman 2016-07-02 05:53

For base 26:

[CODE](26^75993+1)^2-2 is prime! (15529.3052s+0.0087s)[/CODE]

215057 digits.

 rogue 2016-07-06 22:24

(2^688042-1)^2-2 is prime at 414243 digits. This is a new record for that form and will enter the top 5000 in the 2700 range.

 paulunderwood 2016-07-06 22:37

[QUOTE=rogue;437699](2^688042-1)^2-2 is prime at 414243 digits. This is a new record for that form and will enter the top 5000 in the 2700 range.[/QUOTE]

Congrats for another one! :smile:

 rogue 2016-07-17 02:10

I have been very lucky.

(2^695631-1)^2-2 is prime at just over 418,800 digits. It will be around 2600 in the Top 5000.

 henryzz 2016-07-27 08:04

For base 24:
[CODE](24^20047-1)^2-2 is 3-PRP![/CODE]

 BotXXX 2016-11-24 17:31

Following [url]http://www.mersenneforum.org/showpost.php?p=447704&postcount=69[/url]

(2010^3+1)^2-2 is 3-PRP! (0.0000s+0.0001s)
Primality testing (2010^3+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 1+sqrt(13)
Running N+1 test using discriminant 13, base 2+sqrt(13)
(2010^3+1)^2-2 is prime! (0.0227s+0.0010s)

(2010^3-1)^2-2 is 3-PRP! (0.0000s+0.0006s)
Primality testing (2010^3-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N+1 test using discriminant 17, base 1+sqrt(17)
Running N+1 test using discriminant 17, base 2+sqrt(17)
(2010^3-1)^2-2 is prime! (0.0242s+0.0014s)

(2010^35-1)^2-2 is 3-PRP! (0.0018s+0.0001s)
Primality testing (2010^35-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N+1 test using discriminant 23, base 1+sqrt(23)
(2010^35-1)^2-2 is prime! (0.0311s+0.0004s)

(2010^1967-1)^2-2 is 3-PRP! (3.4481s+0.0003s)
Primality testing (2010^1967-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N+1 test using discriminant 29, base 1+sqrt(29)
Running N+1 test using discriminant 29, base 2+sqrt(29)
(2010^1967-1)^2-2 is prime! (32.5536s+0.0007s)

OpenPFGW 3.8.0 on Windows with an Intel Core i7-3667U
Just to get a feeling of cksieve and openpwfg (a long long time ago), after the initial start sieve (-P1e9) I checked 1 <= n <= 2000. The above are the 4 primes found. Will continue to sieve and test this base.

To keep the format of [url]http://www.mersenneforum.org/rogue/ckps.html[/url] :
base 2010 (-1) 3 35 1967
base 2010 (+1) 3

 BotXXX 2017-01-20 16:10

(2010^30505+1)^2-2 is 3-PRP! (1158.2858s+0.0108s)
Primality testing (2010^30505+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 17, base 1+sqrt(17)
(2010^30505+1)^2-2 is prime! (7971.5683s+0.0110s)

201528 digits - ie too small for the top 5 .. ;)

 BotXXX 2017-03-06 10:08

Current primes (for both bases upto n = 40.000)

base 316 (-1) 1 5 27 183 5331 14854 17396
base 316 (+1) 9 41 360 521 6421

base 2010 (-1) 3 35 1967
base 2010 (+1) 3 30505

(316^1-1)^2-2 is 3-PRP! (0.0000s+0.0001s)
Primality testing (316^1-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
(316^1-1)^2-2 is prime! (0.0001s+0.0008s)

(316^5-1)^2-2 is 3-PRP! (0.0000s+0.0007s)
Primality testing (316^5-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 17, base 1+sqrt(17)
Running N+1 test using discriminant 17, base 2+sqrt(17)
(316^5-1)^2-2 is prime! (0.0283s+0.0004s)

(316^9+1)^2-2 is 3-PRP! (0.0000s+0.0001s)
Primality testing (316^9+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Running N+1 test using discriminant 13, base 2+sqrt(13)
(316^9+1)^2-2 is prime! (0.0276s+0.0006s)

(316^27-1)^2-2 is 3-PRP! (0.0003s+0.0002s)
Primality testing (316^27-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 2+sqrt(13)
Running N+1 test using discriminant 13, base 3+sqrt(13)
(316^27-1)^2-2 is prime! (0.0456s+0.0004s)

(316^41+1)^2-2 is 3-PRP! (0.0008s+0.0003s)
Primality testing (316^41+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Running N+1 test using discriminant 13, base 2+sqrt(13)
(316^41+1)^2-2 is prime! (0.0338s+0.0004s)

(316^183-1)^2-2 is 3-PRP! (0.0147s+0.0001s)
Primality testing (316^183-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 2+sqrt(11)
Running N+1 test using discriminant 11, base 3+sqrt(11)
(316^183-1)^2-2 is prime! (0.1514s+0.0004s)

(316^360+1)^2-2 is 3-PRP! (0.0625s+0.0002s)
Primality testing (316^360+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Running N+1 test using discriminant 13, base 2+sqrt(13)
(316^360+1)^2-2 is prime! (0.4942s+0.0004s)

(316^521+1)^2-2 is 3-PRP! (0.1391s+0.0002s)
Primality testing (316^521+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Running N+1 test using discriminant 13, base 2+sqrt(13)
(316^521+1)^2-2 is prime! (1.0966s+0.0004s)

(316^5331-1)^2-2 is 3-PRP! (18.1007s+0.0008s)
Primality testing (316^5331-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 2+sqrt(11)
Running N+1 test using discriminant 11, base 3+sqrt(11)
(316^5331-1)^2-2 is prime! (148.4876s+0.0012s)

(316^6421+1)^2-2 is 3-PRP! (28.4269s+0.0010s)
Primality testing (316^6421+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 29, base 2+sqrt(29)
Running N+1 test using discriminant 29, base 4+sqrt(29)
(316^6421+1)^2-2 is prime! (261.8544s+0.0013s)

(316^14854-1)^2-2 is 3-PRP! (152.9047s+0.0040s)
Primality testing (316^14854-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 17, base 1+sqrt(17)
Running N+1 test using discriminant 17, base 2+sqrt(17)
(316^14854-1)^2-2 is prime! (1238.4924s+0.0034s)

(316^17396-1)^2-2 is 3-PRP! (185.3647s+0.0047s)
Primality testing (316^17396-1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 13, base 2+sqrt(13)
Running N+1 test using discriminant 13, base 3+sqrt(13)
(316^17396-1)^2-2 is prime! (1956.7175s+0.0038s)

 Dylan14 2017-04-16 01:30

For base 50, 1 <= n <= 50000, I found the following primes:
[CODE] [FONT=Times New Roman, serif](50^1-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^3-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^4+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^4-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^9-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^31-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^38+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^66-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^93+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^115-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^120+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^430-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^1233-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^2546-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^2674-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^4396+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^6360-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^11459+1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^25887+1)^2-2[/FONT]
[/CODE]

 sweety439 2017-04-18 18:40

I searched bases 42, 44, 46 and 48 and found those primes: (for n>100)

(42^195-1)^2-2
(42^255-1)^2-2
(42^713-1)^2-2
(42^119+1)^2-2
(44^1288-1)^2-2
(44^195+1)^2-2
(44^1482+1)^2-2
(46^269-1)^2-2
(46^1304-1)^2-2
(48^207+1)^2-2
(48^329+1)^2-2
(48^1153+1)^2-2

Continue searching...

 sweety439 2017-04-19 19:17

Found another prime:

(44^1947-1)^2-2

 rogue 2017-04-22 18:44

sweety439, what range of n are you searching?

 sweety439 2017-04-23 14:04

I tested to about n=2000 and found only these primes. At beginning, I decided to search to n=10K, but it will take too much time, so I released it.

 rogue 2017-04-23 15:47

[QUOTE=sweety439;457323]I tested to about n=2000 and found only these primes. At beginning, I decided to search to n=10K, but it will take too much time, so I released it.[/QUOTE]

If you properly sieved, it should take less than a day to test n=10000.

 kar_bon 2017-04-24 08:55

Primes for CK Base 44, tested up to n=25K:

(44^8210+1)^2-2
(44^12909-1)^2-2
(44^20502+1)^2-2

 vasyannyasha 2017-04-28 08:31

base 252 up to n=2000
(252^88-1)^2-2
(252^177+1)^2-2
(252^337-1)^2-2
(252^717-1)^2-2 is 3-PRP! (0.3107s+0.0002s)
(252^1330-1)^2-2 is 3-PRP! (0.9531s+0.0002s)
(252^1468+1)^2-2 is 3-PRP! (1.6530s+0.0002s)
(252^1996-1)^2-2 is 3-PRP! (2.4196s+0.0003s)
if program write ''(252^1996-1)^2-2 is 3-PRP!'' it means that this number really prime or 50/50?
Pre-thanks
P.S first three really primes.

 kar_bon 2017-04-28 09:02

With pfgw use the "-tp" option to prove prime.

So
pfgw -q"(252^717-1)^2-2"

will show
(252^717-1)^2-2 is 3-PRP!

but
pfgw64 -tp -q"(252^717-1)^2-2"

will show
Primality testing (252^717-1)^2-2 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 17, base 1+sqrt(17)
(252^717-1)^2-2 is prime! (0.6952s+0.0011s)

 rogue 2017-04-28 14:26

FYI, you only need to report in increments of 10,000. It will save me time.

 vasyannyasha 2017-04-29 14:48

[QUOTE=rogue;457782]FYI, you only need to report in increments of 10,000. It will save me time.[/QUOTE]
Ok.
Finally for base 252 from n=1 to n=10000 i find 9 primes(7 Carol primes and 2 Kynea primes).
(252^88-1)^2-2 is prime! (0.0380s+0.0005s)
(252^177+1)^2-2 is prime! (0.0915s+0.0009s)
(252^337-1)^2-2 is prime! (0.2384s+0.0009s)
(252^717-1)^2-2 is prime! (1.2084s+0.0009s)
(252^1330-1)^2-2 is prime! (8.1088s+0.0236s)
(252^1468+1)^2-2 is prime! (8.6567s+0.0053s)
(252^1996-1)^2-2 is prime! (16.5803s+0.0061s)
(252^3095-1)^2-2 is prime! (37.3119s+0.0015s)
(252^6548-1)^2-2 is prime! (159.6788s+0.0013s)

 Dylan14 2017-06-27 16:04

For base 50, 50001 <= n <= 100000, the following primes were found:

[CODE] [FONT=Times New Roman, serif](50^53351-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^69033-1)^2-2[/FONT]
[FONT=Times New Roman, serif](50^69157-1)^2-2[/FONT]
[/CODE]

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