A005574 -id:A005574 - cp's OEIS frontend

A056994 Numbers k such that k^128 + 1 is prime.

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

Offset: 1

Robert G. Wilson v, Sep 06 2000

nonn

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Yves Gallot, Generalized Fermat Prime Search.

Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

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} ]

isA056994(n) = isprime(n^128+1) \\ Michael B. Porter, Mar 30 2010

A006315 Numbers n such that n^32 + 1 is prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005574, A000068, A006314, A006313, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

[n: n in [1..3000] | IsPrime(n^32 + 1)]; // Vincenzo Librandi, Sep 25 2012

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 *)

isA006315(n) = isprime(n^32+1) \\ Michael B. Porter, Mar 26 2010

More terms from Hugo Pfoertner, Jun 22 2003

A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444

Offset: 0

Robert G. Wilson v, Sep 06 2000

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

			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, ....

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

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).

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 *)

a(n)=my(k=2);while(!isprime(k^(2^n)+1),k++);k \\ Anders Hellström, Sep 16 2015

a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022

from Robert G. Wilson v, Oct 30 2000
from Jeppe Stig Nielsen, Aug 07 2005
and 75898 from Lei Zhou, Feb 01 2012
919444 from Jeppe Stig Nielsen, Dec 30 2017

A006316 Numbers k such that k^64 + 1 is prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

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).

Simon Plouffe, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005574, A000068, A006314, A006313, A006315, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

[n: n in [1..4200] | IsPrime(n^64 + 1)]; // Vincenzo Librandi, Sep 25 2012

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 *)

isA006316(n) = isprime(n^64+1) \\ Michael B. Porter, Mar 28 2010

More terms from Hugo Pfoertner, Jun 22 2003

A057002 Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 09 2000

nonn

This sequence is infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, Apr 26 2012

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

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.

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 *)

isA057002(n) = isprime(n^1024+1) \\ Michael B. Porter, Apr 03 2010

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

A056995 Numbers k such that k^256 + 1 is prime.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 06 2000

nonn

Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.

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

Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

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} ]

isA056995(n) = isprime(n^256+1) \\ Michael B. Porter, Apr 01 2010

A057465 Numbers k such that k^512 + 1 is prime.

Original entry on oeis.org

1, 46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, 8678, 8792, 9448, 9452, 9972, 10086, 10448, 10926, 11468, 12754, 13198, 13776, 14734, 16826, 16914, 18334

Offset: 1

Robert G. Wilson v, Sep 08 2000

nonn

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.

T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search.

Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Do[ If[ PrimeQ[ n^512 + 1 ], Print[ n ] ], {n, 0, 31269} ]

isA057465(n) = isprime(n^512+1) \\ Michael B. Porter, Apr 02 2010

A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, 110540, 114690, 125440, 125442, 127596, 138068, 144362, 154908, 157310, 161822, 161900, 166224

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Yves Gallot, Generalized Fermat Prime Search.
Jeppe Stig Nielsen, All terms up to 2*10^9 (from a distributed search)
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
'stream', GFN-12 Final Statistic.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088362(n)=isprime(n^4096+1) \\ Michael B. Porter, Apr 22 2010

A088361 Numbers n such that n^2048 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, 46502, 47348, 49190, 49204, 49544, 54514, 57210, 59770, 61184, 66894, 68194, 70574, 72446, 82642

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

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.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088361(n)=isprime(n^2048+1) \\ Michael B. Porter, Apr 23 2010

A049422 Numbers k such that k^2 + 3 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 22, 28, 38, 50, 52, 62, 64, 70, 74, 76, 92, 94, 106, 112, 122, 130, 134, 140, 146, 154, 158, 160, 172, 178, 218, 230, 242, 244, 248, 256, 274, 286, 298, 304, 316, 322, 326, 340, 350, 356, 364, 368, 398, 406, 416, 424, 430, 434, 440, 458, 470

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

			4^2 + 3 = 19, which is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), this sequence (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).

lst={};Do[If[PrimeQ[n^2+3], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

isA049422(n) = isprime(n^2+3) \\ Michael B. Porter, Mar 19 2010

cp's OEIS Frontend

A056994 Numbers k such that k^128 + 1 is prime.

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A006315 Numbers n such that n^32 + 1 is prime.

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A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

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A006316 Numbers k such that k^64 + 1 is prime.

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A057002 Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

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A056995 Numbers k such that k^256 + 1 is prime.

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A057465 Numbers k such that k^512 + 1 is prime.

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A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).