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Search: a253854 -id:a253854
Displaying 1-10 of 21 results found.
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A005574
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
Also, nonnegative integers such that a(n)+i is a Gaussian prime. - Maciej Ireneusz Wilczynski, May 30 2011
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
T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
F. Ellermann, Primes of the form (m^2)+1 up to 10^6.
L. Euler, Lettre CXLIX (to Goldbach), 1752.
L. Euler, De numeris primis valde magnis, Novi Commentarii academiae scientiarum Petropolitanae 9 (1764), pp. 99-153. See pp. 123-125.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
F. Lemmermeyer, Primes of the form a^2+1, Math Overflow question (2010).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Power.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.
FORMULA
a(n) = A090693(n) - 1.
a(n) = 2*A001912(n-1) for n > 1. - Jeppe Stig Nielsen, Aug 04 2019
MATHEMATICA
Select[Range[350], PrimeQ[ #^2 + 1] &] (* Stefan Steinerberger, Apr 06 2006 *)
Join[{1}, 2Flatten[Position[PrimeQ[Table[x^2+1, {x, 2, 1000, 2}]], True]]] (* Fred Patrick Doty, Aug 18 2017 *)
PROG
(PARI) isA005574(n) = isprime(n^2+1) \\ Michael B. Porter, Mar 20 2010
(PARI) for(n=1, 1e3, if(isprime(n^2 + 1), print1(n, ", "))) \\ Altug Alkan, Oct 14 2015
(Magma) [n: n in [0..400] | IsPrime(n^2+1)]; // Vincenzo Librandi, Nov 18 2010
(Haskell)
a005574 n = a005574_list !! (n-1)
a005574_list = filter ((== 1) . a010051' . (+ 1) . (^ 2)) [0..]
-- Reinhard Zumkeller, Jul 03 2015
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).
Cf. A010051, A259645, A295405 (characteristic function).
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A000068
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
M. Lal, Primes of the form n^4 + 1, Math. Comp., 21 (1967), 245-247.
D. Shanks, On numbers of the form n^4+1, Math. Comp. 15 (74) (1961), 186-189.
MATHEMATICA
Select[Range[10^2*2], PrimeQ[ #^4+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
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
(Magma) [n: n in [0..800] | IsPrime(n^4+1)]; // Vincenzo Librandi, Nov 18 2010
CROSSREFS
Cf. A002523, A037896, A005574, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A006313
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
MATHEMATICA
lst={}; Do[If[PrimeQ[n^16+1], AppendTo[lst, n]], {n, 10^4}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 20 2008 *)
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 *)
PROG
(PARI) isA006313(n) = isprime(n^16+1) \\ Michael B. Porter, Mar 25 2010
(Magma) [ n: n in [0..1500] | IsPrime(n^16+1) ]; // Vincenzo Librandi, Nov 18 2010
CROSSREFS
Cf. A005574, A000068, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Hugo Pfoertner, Jun 22 2003
STATUS
approved
A006314
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000
MATHEMATICA
Select[Range[1300], PrimeQ[#^8+1]&] (* Harvey P. Dale, Mar 31 2011 *)
PROG
(PARI) isA006314(n) = isprime(n^8+1) \\ Michael B. Porter, Mar 24 2010
(Magma) [n: n in [0..2000] | IsPrime(n^8+1)]; Vincenzo Librandi, Nov 18 2010
CROSSREFS
Cf. A005574, A000068, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from James A. Sellers, May 03 2000
STATUS
approved
A056994
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Yves Gallot, Generalized Fermat Prime Search.
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} ]
PROG
(PARI) isA056994(n) = isprime(n^128+1) \\ Michael B. Porter, Mar 30 2010
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Sep 06 2000
STATUS
approved
A006315
Numbers n such that n^32 + 1 is prime.
(Formerly M5208)
+10
37
1, 30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, 1596, 1678, 1714, 1754, 1812, 1818, 1878, 1906, 1960, 1962, 2046, 2098, 2118, 2142, 2330, 2418, 2434, 2654, 2668
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
MATHEMATICA
lst={}; Do[If[PrimeQ[n^32+1], Print[n]; AppendTo[lst, n]], {n, 10^5}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Range[0, 2700], PrimeQ[(#^32 + 1)] &] (* Vincenzo Librandi, Sep 25 2012 *)
PROG
(PARI) isA006315(n) = isprime(n^32+1) \\ Michael B. Porter, Mar 26 2010
(Magma)[n: n in [1..3000] | IsPrime(n^32 + 1)]; // Vincenzo Librandi, Sep 25 2012
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Hugo Pfoertner, Jun 22 2003
STATUS
approved
A056993
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
LINKS
Table of n, a(n) for n=0..20.
Yves Gallot, Generalized Fermat Prime Search
Lucile and Yves Gallot, Generalized Fermat Prime Search
Michael Goetz, id=103235 of Top 5000 Primes
Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption, Cryptology ePrint Archive 2023/700.
Stephen Scott, id=84401 of Top 5000 Primes
Sylvanus A. Zimmerman, PrimeGrid’s Generalized Fermat Prime Search
FORMULA
a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022
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.
Cf. A019434 (Fermat primes).
KEYWORD
hard,more,nonn
AUTHOR
Robert G. Wilson v, Sep 06 2000
EXTENSIONS
1534 from Robert G. Wilson v, Oct 30 2000
62722 from Jeppe Stig Nielsen, Aug 07 2005
24518 and 75898 from Lei Zhou, Feb 01 2012
919444 from Jeppe Stig Nielsen, Dec 30 2017
STATUS
approved
A006316
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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).
LINKS
Simon Plouffe, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
MATHEMATICA
lst={}; Do[If[PrimeQ[n^64+1], Print[n]; AppendTo[lst, n]], {n, 10^4}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Range[0, 4200], PrimeQ[(#^64 + 1)] &] (* Vincenzo Librandi, Sep 25 2012 *)
PROG
(PARI) isA006316(n) = isprime(n^64+1) \\ Michael B. Porter, Mar 28 2010
(Magma) [n: n in [1..4200] | IsPrime(n^64 + 1)]; // Vincenzo Librandi, Sep 25 2012
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Robert G. Wilson v
EXTENSIONS
More terms from Hugo Pfoertner, Jun 22 2003
STATUS
approved
A057002
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, Apr 26 2012
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number
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 *)
PROG
(PARI) isA057002(n) = isprime(n^1024+1) \\ Michael B. Porter, Apr 03 2010
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.
Cf. A006093.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Sep 09 2000
EXTENSIONS
More terms from Jeppe Stig Nielsen, Sep 27 2003
Edited at the suggestion of T. D. Noe by N. J. A. Sloane, May 14 2008
STATUS
approved
A056995
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
LINKS
Simon Plouffe, Table of n, a(n) for n = 1..10000 (1000 first terms from T. D. Noe)
Yves Gallot, Generalized Fermat Prime Search.
Simon Plouffe, 146309 terms
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} ]
PROG
(PARI) isA056995(n) = isprime(n^256+1) \\ Michael B. Porter, Apr 01 2010
CROSSREFS
Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Sep 06 2000
STATUS
approved
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