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A098845 Numbers k such that 4^k - 2^k - 1 is prime. 10
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)