%I #64 Apr 03 2023 10:36:10

%S 2,4,5,9,10,18,38,45,50,57,108,161,208,224,225,240,354,597,634,1008,

%T 1080,1468,1525,1560,3298,3329,3846,4129,5430,8616,11834,12988,14610,

%U 43401,45306,53776,54449,67497,74025,122449,136845,142896,164541,171157,187668,274054,316944,349296

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

%C All primes certified using PFGW from primeform group. - _Pierre CAMI_, Mar 07 2005

%C 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

%C 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

%H Chris Caldwell, <a href="https://t5k.org/primes/page.php?id=80257">The largest known primes</a>

%H Mike Hamburg, <a href="https://eprint.iacr.org/2015/625">Ed448-Goldilocks, a new elliptic curve</a>, Cryptology ePrint Archive, Report 2015/625.

%p select(t -> isprime(4^t-2^t-1), [$1..1000]); # _Robert Israel_, Dec 08 2015

%t Select[Range[15000], PrimeQ[4^# - 2^# - 1] &] (* _Vincenzo Librandi_, Dec 08 2015 *)

%o (Magma) [n: n in [0..1000] | IsPrime(2^n*(2^n-1)-1)]; // _Vincenzo Librandi_, Dec 08 2015

%o (PARI) for(n=1, 1e3, if(ispseudoprime(4^n-2^n-1), print1(n, ", "))) \\ _Altug Alkan_, Dec 08 2015

%o (Python)

%o from sympy import isprime

%o for n in range(1,1000):

%o if isprime(4**n-2**n-1):

%o print(n, end=', ') # _Stefano Spezia_, Jan 11 2019

%Y Cf. similar sequences listed in A265481.

%K nonn

%O 1,1

%A _Pierre CAMI_, Oct 10 2004; extended several times: Jun 01 2005, Jun 19 2006, May 03 2007

%E Extended to a(44) = 349296 (2^698592 - 2^349296 - 1 is a 210298-digit certified prime) by _Pierre CAMI_, Jan 11 2009

%E Definition simplified by _Pierre CAMI_, May 10 2012

%E a(30) corrected by _Robert Israel_, Dec 14 2015

%E 4 missing terms between a(41) = 136845 and what is now a(46) = 274054 added by _Fabrice Lavier_, Jan 10 2019