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