A157950 -id:A157950 - cp's OEIS frontend

A158361 Primes p with property that Q = p^4 + 2^4 is prime.

3, 5, 7, 11, 17, 19, 23, 37, 41, 59, 61, 71, 79, 97, 131, 139, 179, 223, 227, 229, 241, 283, 313, 317, 359, 367, 379, 383, 389, 439, 449, 461, 487, 503, 521, 569, 593, 617, 619, 631, 661, 683, 709, 733, 811, 821, 853, 911, 977, 1049, 1061, 1063, 1069, 1091, 1093, 1117

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2009

Q is always congruent to 1 (mod 4).
Q is divisible by 17 if p is congruent to 1, 4, 13, or 16 (mod 17).
It is conjectured that sequence a(n) is infinite.
Q is in A094479. - Zak Seidov, Jul 08 2020

			3 is in the sequence since for p=3: p^4+2^4 = 3^4+16 = 97 is prime.
29 is not in the sequence since 29^4+2^4 = 707297 = 73 x 9689 is not prime.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Richard Guy, "Unsolved Problems in Number Theory"

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062324, A094479, A157950, A157764.

[p: p in PrimesUpTo(2000) | IsPrime(p^4+16)]; // Vincenzo Librandi, Jun 18 2014

Select[Range[10^3], PrimeQ[#] && PrimeQ[#^4 + 16] &] (* Vincenzo Librandi, Jun 18 2014 *)
Select[Prime[Range[200]],PrimeQ[#^4+16]&] (* Harvey P. Dale, Jun 23 2014 *)

isA158361(n) = isprime(n) && isprime(n^4+16)

Corrected and edited by Michael B. Porter, Dec 17 2009

A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

Original entry on oeis.org

13, 89, 137, 29, 107, 223, 37, 59, 127, 331, 113, 53, 101, 139, 389, 113, 223, 181, 103, 173, 491, 13, 1223, 5279, 491, 109, 179, 563, 1151, 181, 1277, 7517, 547, 181, 229, 647, 43, 2153, 761, 1993, 8039, 619, 199, 233, 701, 53, 271, 3559, 4133, 2399, 9833, 661, 379, 349, 773

Offset: 3

Jonathan Vos Post, Aug 15 2007

These were computed by Ignacio Larrosa Cañestro, who cautions that some are only probable primes. The k=3 row is A157950. The main diagonal is A132261.

			The array begins:
   n  |   1    2    3    4    5    6     7     8     9    10
  ----+--------------------------------------------------------
  k=3 |  13  137  223  331  389  491   563   647   701   773
  k=4 |  89  107  127  139  173  179   229   233   349   421
  k=5 |  29   59  101  103  109  181   199   379   769   881
  k=6 |  37   53  181  491  547  619   661   677   911   941
  k=7 | 113  223 5279 7517 8039 9833 12197 13757 21467 23447
  k=8 | 113 1223 1277 1993 2399 9349  9739 10211 10973 11059

Cf. A000040, A062324, A132261, A157764, A157950, A158361, A158477.

More terms from Jinyuan Wang, Feb 01 2022

A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

Original entry on oeis.org

29, 59, 101, 103, 109, 181, 199, 379, 769, 881, 919, 977, 1097, 1213, 1303, 1583, 2099, 2113, 2441, 2521, 2617, 2777, 3067, 3739, 4133, 4289, 4519, 4931, 5039, 5113, 5227, 5417, 5743, 5783, 6143, 6373, 6691, 8053, 8209, 8287, 8513, 9109, 9203, 9689, 9787, 9923, 9941

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2009

nonn

1) Q=(p^16)^2+(2^16)^2 only for Q=4k+1 because of Fermat/Euler/Lagrange theorem concerning prime as sum of two squares.
2) It is conjectured that sequence a(n) is infinite.
3) Note the twin prime: a(3)=101, a(4)=103.
The next set of twins is a(101)=30557, a(102)=30559. - Robert Israel, Apr 05 2016

			p=3: 3^32+2^32=1853024483819137 = 1153 x 1607133116929 no prime;
also for following primes p=5, 7, 11, 13, 17, 19, 23: Q(p) no prime;
p=29: 29^32+2^32=62623297589448778360828428329074752313100292737 is prime => a(1)=29.

Richard E. Crandall, Carl Pomerance, Prime Numbers: A Computational Perspective, Springer 2001.
Leonard E. Dickson, History of the Theory of Numbers, Dover Pub. Inc., 2005.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062324, A157764, A157950, A158361.

select(t -> isprime(t) and isprime(t^32 + 2^32), [seq(i,i=3..10000,2)]); # Robert Israel, Apr 05 2016

With[{c=2^32},Select[Prime[Range[1300]],PrimeQ[#^32+c]&]] (* Harvey P. Dale, May 04 2018 *)

isA158477(n) = isprime(n) && isprime(n^32+4294967296) \\ Michael B. Porter, Dec 17 2009

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p^32+2^32), print1(p, ", "))); \\ Altug Alkan, Apr 05 2016

n^32+2^32 and n to be prime.

cp's OEIS Frontend

A158361 Primes p with property that Q = p^4 + 2^4 is prime.

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A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

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A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

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Maple

Mathematica

PARI

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