A098845 Numbers k such that 4^k - 2^k - 1 is prime.

2, 4, 5, 9, 10, 18, 38, 45, 50, 57, 108, 161, 208, 224, 225, 240, 354, 597, 634, 1008, 1080, 1468, 1525, 1560, 3298, 3329, 3846, 4129, 5430, 8616, 11834, 12988, 14610, 43401, 45306, 53776, 54449, 67497, 74025, 122449, 136845, 142896, 164541, 171157, 187668, 274054, 316944, 349296

Offset

1,1

Comments

All primes certified using PFGW from primeform group. - Pierre CAMI, Mar 07 2005

No terms 2, 3, 7, 12, 13 or 15 (mod 20) except 2. - Robert Israel, Dec 08 2015, updated by Fabrice Lavier, Jan 10 2019

Using such "Goldilocks" primes (a term coined by Mike Hamburg) as modulus facilitates use of Karatsuba multiplication in elliptic-curve cryptography. - Francois R. Grieu, Mar 25 2021

Links

Table of n, a(n) for n=1..48.

Chris Caldwell, The largest known primes

Mike Hamburg, Ed448-Goldilocks, a new elliptic curve, Cryptology ePrint Archive, Report 2015/625.

Maple

select(t -> isprime(4^t-2^t-1), [$1..1000]); # Robert Israel, Dec 08 2015

Mathematica

Select[Range[15000], PrimeQ[4^# - 2^# - 1] &] (* Vincenzo Librandi, Dec 08 2015 *)

Prog

(Magma) [n: n in [0..1000] | IsPrime(2^n*(2^n-1)-1)]; // Vincenzo Librandi, Dec 08 2015
(PARI) for(n=1, 1e3, if(ispseudoprime(4^n-2^n-1), print1(n, ", "))) \\ Altug Alkan, Dec 08 2015
(Python)
from sympy import isprime
for n in range(1, 1000):
    if isprime(4**n-2**n-1):
        print(n, end=', ') # Stefano Spezia, Jan 11 2019

Crossrefs

Cf. similar sequences listed in A265481.

Sequence in context: A319423 A265748 A191001 * A298981 A069001 A287181

Adjacent sequences: A098842 A098843 A098844 * A098846 A098847 A098848

Keyword

nonn

Author

Pierre CAMI, Oct 10 2004; extended several times: Jun 01 2005, Jun 19 2006, May 03 2007

Extensions

Extended to a(44) = 349296 (2^698592 - 2^349296 - 1 is a 210298-digit certified prime) by Pierre CAMI, Jan 11 2009

Definition simplified by Pierre CAMI, May 10 2012

a(30) corrected by Robert Israel, Dec 14 2015

4 missing terms between a(41) = 136845 and what is now a(46) = 274054 added by Fabrice Lavier, Jan 10 2019

Status

approved