A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163

Offset: 1

Author

Zak Seidov, Aug 28 2014

Keywords

• nonn

Comments

For even k > 2, 4 + n^k is prime only for n = 1.

From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)

4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.

4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.

(End)

All terms are == {3,7} (mod 10). - Zak Seidov, Aug 29 2014

Examples

			From _K. D. Bajpai_, Jan 20 2015: (Start)
a(2) = 5503:
4 + 5503 = 5507;
4 + 5503^2 = 30283013;
4 + 5503^3 = 166647398531;
4 + 5503^5 = 5046584669419727747;
all five are prime.
(End)
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007591, A073573, A125260, A172367.

Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937. - K. D. Bajpai, Jan 20 2015

The subsequence with 4+p^7 also prime is A246562. - Danny Rorabaugh, Apr 19 2015

Programs

Magma

[p: p in PrimesUpTo(2*10^7) | IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015

Mathematica

k=4; Select[Prime[Range[1,500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&]  (*K. D. Bajpai, Jan 20 2015 *)

PARI

for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014

PARI

p=7; forprime(q=11, 1e8, if(q-p==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014

Python

from sympy import prime, isprime
A246519_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(4+p**z) for z in (1,2,3,5)])]
# Chai Wah Wu, Sep 08 2014