A274519 Numbers k for which 4^k - 27 is prime.

3, 4, 5, 10, 11, 13, 25, 28, 29, 65, 70, 115, 305, 515, 2029, 2393, 2605, 3530, 4036, 4750, 10288, 11048, 11596, 29359, 32123, 47371

Offset: 1

Author

Timothy L. Tiffin, Aug 07 2016

Keywords

• nonn
• more

Comments

The prime numbers that these exponents generate are given in A275750.

Since 4^(6k) - 27 = 4096^k - 27 == (1^k - 27) mod 13 = -26 mod 13 == 0 mod 13, no multiple of 6 will be in this sequence. Also, since 4^(5k+2) - 27 = 16*1024^k - 27 == (16*1^k - 27) mod 11 = -11 mod 11 == 0 mod 11, no number congruent to 2 mod 5 will be in this sequence.

a(21) > 5000. - Vincenzo Librandi, Aug 08 2016

Examples

			a(1) = 3, since 4^3 - 27 = 64 - 27 = 37, which is prime.
a(2) = 4, since 4^4 - 27 = 256 - 27 = 229, which is prime.
a(3) = 5, since 4^5 - 27 = 1024 - 27 = 997, which is prime.
a(4) = 10, since 4^10 - 27 = 1048576 - 27 = 1048549, which is prime.
a(5) = 11, since 4^11 - 27 = 4194304 - 27 = 4194277, which is prime.
a(6) = 13, since 4^13 - 27 = 67108864 - 27 = 67108837, which is prime.
		

Links

• D. Alpern, Factorization using the Elliptic Curve Method.

Crossrefs

Cf. A275750, A275767.

Programs

Magma

[n: n in [3..2000] |IsPrime(4^n-27)]; // Vincenzo Librandi, Aug 08 2016

Mathematica

Select[Range[3, 5000], PrimeQ[4^# - 27] &] (* Vincenzo Librandi, Aug 08 2016 *)

Python

from sympy import isprime
def afind(limit, startk=3):
    alst, pow4 = [], 4**startk
    for k in range(startk, limit+1):
        if isprime(pow4 - 27): print(k, end=", ")
        pow4 *= 4
afind(600) # Michael S. Branicky, Sep 22 2021

Extensions

a(13)-a(20) from Vincenzo Librandi, Aug 08 2016

a(21)-a(23) from Michael S. Branicky, Sep 22 2021

a(24)-a(26) from Michael S. Branicky, Oct 25 2024