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

Author

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

Keywords

• nonn

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

• Chris Caldwell, The largest known primes
• Mike Hamburg, Ed448-Goldilocks, a new elliptic curve, Cryptology ePrint Archive, Report 2015/625.

Crossrefs

Cf. similar sequences listed in A265481.

Programs

Magma

[n: n in [0..1000] | IsPrime(2^n*(2^n-1)-1)]; // Vincenzo Librandi, Dec 08 2015

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 *)

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

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