Displaying 1-10 of 36 results found.
Numbers k such that k^2 + 1 is prime.
(Formerly M1010)
+10
175
1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396
COMMENTS
Hardy and Littlewood conjectured that the asymptotic number of elements in this sequence not exceeding n is approximately c*sqrt(n)/log(n) for some constant c. - Stefan Steinerberger, Apr 06 2006 Apparently Goldbach conjectured that any a > 1 from this sequence can be written as a=b+c where b and c are in this sequence (Lemmermeyer link below). - Jeppe Stig Nielsen, Oct 14 2015 No term > 2 can be both in this sequence and in A001105 because of the Aurifeuillean factorization (2*k^2)^2 + 1 = (2*k^2 - 2*k + 1) * (2*k^2 + 2*k + 1). - Jeppe Stig Nielsen, Aug 04 2019
REFERENCES
Harvey Dubner, "Generalized Fermat primes", J. Recreational Math., 18 (1985): 279-280.
R. K. Guy, "Unsolved Problems in Number Theory", 3rd edition, A2.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 15, Thm. 17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Eric Weisstein's World of Mathematics, Power.
MATHEMATICA
Join[{1}, 2Flatten[Position[PrimeQ[Table[x^2+1, {x, 2, 1000, 2}]], True]]] (* Fred Patrick Doty, Aug 18 2017 *)
PROG
(PARI) for(n=1, 1e3, if(isprime(n^2 + 1), print1(n, ", "))) \\ Altug Alkan, Oct 14 2015 (Haskell)
a005574 n = a005574_list !! (n-1)
a005574_list = filter ((== 1) . a010051' . (+ 1) . (^ 2)) [0..]
CROSSREFS
Cf. A002522, A001912, A002496, A062325, A090693, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323. Other sequences of the type "Numbers k such that k^2 + i is prime": this sequence (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).
Numbers k such that k^4 + 1 is prime.
(Formerly M1027 N0386)
+10
55
1, 2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, 238, 242, 248, 254, 266, 272, 276, 278, 288, 296, 312, 320, 328, 334, 340, 352, 364, 374, 414, 430, 436, 442, 466
REFERENCES
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
PROG
(PARI) {a(n) = local(m); if( n<1, 0, for(k=1, n, until( isprime(m^4 + 1), m++)); m)};
(PARI) list(lim)=my(v=List([1])); forstep(k=2, lim, 2, if(isprime(k^4+1), listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Mar 31 2022
CROSSREFS
Cf. A002523, A037896, A005574, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Numbers n such that n^16 + 1 is prime.
(Formerly M2164)
+10
43
1, 2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, 686, 688, 690, 736, 774, 776, 778, 790, 830, 832, 834, 846, 900, 916, 946, 956, 972, 982, 984, 1018, 1044, 1078
REFERENCES
Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
s=Reap[Sow[1]; Do[If[PrimeQ[n^16+1], Sow[n]], {n, 2, 40352, 2}]][[2, 1]] (* Zak Seidov, Dec 22 2010 *) Join[{1}, 2*Flatten[Position[Range[2, 1100, 2]^16+1, _?PrimeQ]]] (* Harvey P. Dale, Jun 26 2017 *)
CROSSREFS
Cf. A005574, A000068, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Numbers n such that n^8 + 1 is prime.
(Formerly M1308)
+10
42
1, 2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, 800, 808, 866, 876, 884, 892, 916, 918, 934, 956, 990, 1022, 1028, 1054, 1106, 1120, 1174, 1224, 1232, 1256, 1284
REFERENCES
Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
CROSSREFS
Cf. A005574, A000068, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.
+10
41
2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411
COMMENTS
Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.
MATHEMATICA
Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)
PROG
(PARI) for(n=1, 10^3, if(isprime(polcyclo(10, n)), print1(n, ", "))); \\ Joerg Arndt, Nov 13 2014 (Magma) [n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014
CROSSREFS
Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (10), A162862 (11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).
Numbers k such that k^128 + 1 is prime.
+10
39
1, 120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, 9706, 10238, 10994, 11338, 11432, 11466, 11554, 11778, 12704, 12766, 13082, 13478, 13700
REFERENCES
Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
MATHEMATICA
Do[ k = 1; While[ PowerMod[ n, 128, 2*k*128 + 1 ] != 2*k*128 && k < 10^3, k++ ]; If[ k == 10^3 && PrimeQ[ n^128 + 1 ], Print[ n ] ], {n, 2, 15000, 2} ]
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.
+10
36
2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444
COMMENTS
Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner, Jul 01 2003 The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - Hugo Pfoertner, Jul 01 2003 For any n, a(n+1) >= sqrt(a(n)), because k^(2^(n+1))+1 = (k^2)^(2^n)+1. - Jeppe Stig Nielsen, Sep 16 2015 Does the sequence contain any perfect squares? If a(n) is a perfect square, then a(n+1) = sqrt(a(n)). - Jeppe Stig Nielsen, Sep 16 2015 If for a particular n, a(n) exists, then a(i) exist for all i=0,1,2,...,n. No proof is known that this sequence is infinite. Such a result would clearly imply the infinitude of A002496. - Jeppe Stig Nielsen, Sep 18 2015 919444 is a candidate for a(20). See Zimmermann link. - Serge Batalov, Sep 02 2017 Now PrimeGrid has tested and double checked all b^(2^20) + 1 with b < 919444, so we have proof that a(20) = 919444. - Jeppe Stig Nielsen, Dec 30 2017
EXAMPLE
The primes are 2^(2^0) + 1 = 3, 2^(2^1) + 1 = 5, 2^(2^2) + 1 = 17, 2^(2^3) + 1 = 257, 2^(2^4) + 1 = 65537, 30^(2^5) + 1, 102^(2^6) + 1, ....
MATHEMATICA
f[n_] := (p = 2^n; k = 2; While[cp = k^p + 1; !PrimeQ@cp, k++ ]; k); Do[ Print[{n, f@n}], {n, 0, 17}] (* Lei Zhou, Feb 21 2005 *)
PROG
(PARI) a(n)=my(k=2); while(!isprime(k^(2^n)+1), k++); k \\ Anders Hellström, Sep 16 2015
CROSSREFS
Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
EXTENSIONS
24518 and 75898 from Lei Zhou, Feb 01 2012
Numbers k such that k^64 + 1 is prime.
(Formerly M5368)
+10
35
1, 102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, 2420, 2494, 2524, 2614, 2784, 3024, 3104, 3140, 3164, 3254, 3278, 3628, 3694, 3738, 3750, 4000, 4030, 4058, 4166
REFERENCES
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).
+10
35
1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016
MATHEMATICA
Do[ k = 1; While[ PowerMod[ n, 1024, 2*k*1024 + 1 ] != 2*k*1024 && k < 2*10^6, k++ ]; If[ k == 2*10^6 && PrimeQ[ n^1024 + 1 ], Print[ n ] ], {n, 2, 13954, 2} ]
Do[If[PrimeQ[n^1024 + 1], Print[n], ## &[]], {n, 1, 100}] (* Includes first term and runs faster, Daniel Jolly, Nov 04 2014 *)
CROSSREFS
Other sequences of numbers n such that n^(2^k)+1 is prime for fixed k: A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Numbers k such that k^256 + 1 is prime.
+10
34
1, 278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, 8524, 8644, 8762, 8808, 9024, 9142, 9412, 10892, 12206, 13220, 13222, 13246, 13370, 13738, 14114, 14930
REFERENCES
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
MATHEMATICA
Do[ k = 1; While[ PowerMod[ n, 256, 2*k*256 + 1 ] != 2*k*256 && k < 10^3, k++ ]; If[ k == 10^3 && PrimeQ[ n^256 + 1 ], Print[ n ] ], {n, 2, 15000, 2} ]
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
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