Carol / Kynea Primes

rogue · 2016-04-15, 17:28

I've started this thread to capture Carol / Kynea primes. Use this thread to post reservations.

Please go here 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:
https://en.wikipedia.org/wiki/Carol_number
https://en.wikipedia.org/wiki/Kynea_number

NorbSchneider · 2016-04-16, 22:26

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

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

lalera · 2016-04-18, 22:28

Quote:

Originally Posted by wombatman

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

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

axn · 2016-04-20, 08:30

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

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

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:

Originally Posted by rogue

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

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

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

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

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]

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]

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

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

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

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

2016-04-15, 17:28	#1
rogue "Mark" Apr 2003 Between here and the 3×31×71 Posts	Carol / Kynea Primes I've started this thread to capture Carol / Kynea primes. Use this thread to post reservations. Please go here 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: https://en.wikipedia.org/wiki/Carol_number https://en.wikipedia.org/wiki/Kynea_number Last fiddled with by gd_barnes on 2018-02-21 at 20:48 Reason: admin edit

2016-04-16, 22:26	#2
NorbSchneider "Norbert" Jul 2014 Budapest 112₁₀ Posts	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

2016-04-18, 18:39	#3
wombatman I moo ablest echo power! May 2013 1796₁₀ Posts	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

2016-04-21, 18:08	#9
lalera Jul 2003 2×307 Posts	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 continuing

2016-04-28, 22:36	#10
henryzz Just call me Henry "David" Sep 2007 Liverpool (GMT/BST) 3²×5×7×19 Posts	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

2016-04-29, 00:10	#11
WMHalsdorf Feb 2005 Bristol, CT 3³·19 Posts	(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

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