A128025 Numbers k such that (8^k - 3^k)/5 is prime.

2, 3, 7, 19, 31, 67, 89, 9227, 43891, 854149

Offset: 1

Author

Alexander Adamchuk, Feb 11 2007

Keywords

• hard
• more
• nonn

Comments

All terms are primes.

Verified the first 8 terms in sequence. Also, the next number in the sequence, if one exists is > 43691. - Robert Price, Mar 16 2010

a(10) > 10^5. - Robert Price, Jul 27 2011

a(11) > 10^6. - Jon Grantham, Jul 29 2023

Links

• Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Crossrefs

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Programs

Mathematica

k=5; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

PARI

is(n)=isprime((8^n-3^n)/5) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

9227 from Farideh Firoozbakht, Apr 08 2007

a(9) from Robert Price, Jul 27 2011

a(10) from Jon Grantham, Jul 29 2023