A236173
Primes p such that p^2 - p - 1, p^3 - p - 1 and p^4 - p - 1 are all prime.
Original entry on oeis.org
11, 71, 11621, 28151, 32089, 37501, 39209, 45329, 66161, 76649, 114599, 122131, 136949, 154991, 202999, 228901, 243391, 270269, 296911, 313909, 318679, 333701, 343309, 359291, 369979, 371281, 371981, 373171, 373459
Offset: 1
228901 is prime, 228901^2 - 228901 - 1 is prime, 228901^3 - 228901 - 1 is prime, and 228901^4 - 228901 - 1 is prime. So 228901 is a member of this sequence.
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Select[Prime[Range[32000]],AllTrue[#^{2,3,4}-#-1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 08 2019 *)
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s=[]; forprime(p=2, 400000, if(isprime(p^2-p-1) && isprime(p^3-p-1) && isprime(p^4-p-1), s=concat(s, p))); s \\ Colin Barker, Jan 20 2014
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import sympy
from sympy import isprime
{print(p) for p in range(10**6) if isprime(p) and isprime(p**2-p-1) and isprime(p**3-p-1) and isprime(p**4-p-1)}
A236952
Primes p such that p^4 - p +/- 1 are twin primes.
Original entry on oeis.org
13, 79, 3571, 3739, 5023, 5443, 7459, 7621, 10243, 13339, 14251, 17359, 17551, 17863, 17971, 18061, 19483, 21481, 27631, 32611, 37501, 38821, 48463, 49711, 54709, 56443, 57073, 57751, 69313, 71353, 72883, 74293, 81883, 82051, 84223
Offset: 1
13 is prime and 13^4-13-1 (28547) and 13^4-13+1 (28549) are twin primes. So, 13 is a member of this sequence.
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[p: p in PrimesUpTo(90000) | IsPrime(p^4-p-1) and IsPrime(p^4-p+1)]; // Vincenzo Librandi, Feb 14 2014
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Select[Prime[Range[10000]], PrimeQ[#^4 - # - 1] && PrimeQ[#^4 - # + 1]&] (* Vincenzo Librandi, Feb 14 2014 *)
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import sympy
from sympy import isprime
{print(n) for n in range(10**6) if isprime(n) and isprime(n**4-n-1) and isprime(n**4-n+1)}
A236073
Primes p such that p^4 + p + 1 and p^4 - p - 1 are also prime.
Original entry on oeis.org
2, 5, 11, 239, 1871, 4001, 4397, 6971, 12647, 12689, 13337, 13619, 15401, 19391, 19559, 19739, 20201, 20297, 22871, 22937, 28307, 30029, 32561, 36299, 36929, 39569, 44279, 45497, 47441, 48767, 50069, 53897, 55871
Offset: 1
6971 is prime, 6971^4 - 6971 - 1 is prime, and 6971^4 + 6971 + 1 is prime. So 6971 is a member of this sequence.
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s=[]; forprime(p=2, 55871, if(isprime(p^4+p+1)&&isprime(p^4-p-1), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014
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import sympy
from sympy import isprime
{print(p) for p in range(10**5) if isprime(p**4+p+1) and isprime(p**4-p-1) and isprime(p)}
Showing 1-3 of 3 results.
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