Carol / Kynea Primes
 

I've started this thread to capture Carol / Kynea primes. Use [URL="http://www.mersenneforum.org/showthread.php?t=21216"]this thread[/URL] to post reservations.

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.

Admin edit: Here are links to Wikipedia about these primes:
[url]https://en.wikipedia.org/wiki/Carol_number[/url]
[url]https://en.wikipedia.org/wiki/Kynea_number[/url]

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

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]

[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

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).

[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]

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]

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

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

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]

(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

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

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])

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

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

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=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?

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=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.

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=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.

Will do. Thanks.

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=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.

[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

I see. Fixed.

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

Thanks Serge for the tip!

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

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

Checked to 10k

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

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

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
--

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]

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]

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]

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]

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]

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
--

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.

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]

[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?

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.

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?

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.

[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?

[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.

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.

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

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

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.

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:

[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.

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.

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]

Congrats :banana:

[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).

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

[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.

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.

[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]

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

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

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

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

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

yes

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

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

Arrgh! :chappy:
Still a very decent number!

[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:

(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]

[QUOTE=rogue;432728]I've started this thread to capture Carol / Kynea primes. Use [URL="http://www.mersenneforum.org/showthread.php?t=21216"]this thread[/URL] to post reservations.

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.

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

[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=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:

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

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

Congrats again :smile:

For base 26:

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

Still working to n=100,000.

For base 26:

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

215057 digits.

(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=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:

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.

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

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

(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 .. ;)

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)

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]

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...

Found another prime:

(44^1947-1)^2-2

sweety439, what range of n are you searching?

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=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.

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

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)
please explain
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.

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)

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

[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)

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]