%I #23 Oct 22 2019 04:21:20

%S 5,17,257,65537,1336337,19170731299728100000001,

%T 285347346718226949041792907369577650673693754163660005676181161059099319730177,

%U 29585383599687066848440635756425168157198788892517565295922752892368299949134315777

%N Prime numbers which are successors of a power of a Fibonacci number.

%C Let a(n) = Fibonacci(x)^y+1, then there exists some a,b > 0, such that x = 3*a and y = 2^b. For the example a(5) = 1336337: x = 9, y = 4, a = 3 and b = 2.

%C Last digit seems to be usually 7, except for a(1) and a(6). - _Alexander Adamchuk_, Aug 09 2006

%H U. Dudley, B. Tucker, <a href="https://www.fq.math.ca/Scanned/9-1/dudley.pdf">Greatest Common Divisors in Altered Fibonacci Sequences</a>, Fibonacci Quarterly 1971, pages 89-91.

%H Letter from Toby Gee in Mathematical Spectrum, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol29_No3.pdf">Fibonacci numbers</a>, vol. 29 (1996/1997), page 66.

%H V. E. Hoggatt Jr. and M. Bicknell-Johnson, <a href="https://www.fq.math.ca/Scanned/15-1/hoggatt1.pdf">Composites and Primes Among Powers of Fibonacci Numbers increased or decreased by one</a>, Fibonacci Quarterly vol. 15 (1977), page 2.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#noprimenghbrs">The Mathematical Magic of the Fibonacci Numbers</a>.

%e a(5) = 1336337 because 1336337 is prime, and 1336337-1 = 1336336 = 34^4+1 = Fibonacci(9)^4+1.

%e a(6) = Fibonacci(15)^8 + 1, a(7) = Fibonacci(48)^8 + 1, a(8) = Fibonacci(51)^8 + 1, a(9) = Fibonacci(63)^8 + 1, a(10) = Fibonacci(21)^32 + 1, a(11) = Fibonacci(198)^4 + 1, a(12) = Fibonacci(204)^8 + 1, a(13) = Fibonacci(366)^8 + 1. - _Alexander Adamchuk_, Aug 09 2006

%t Select[Take[Intersection[Flatten[Table[Fibonacci[3n]^(2^m)+1,{n,1,300},{m,1,7}]]],{1,400}],PrimeQ] (* _Alexander Adamchuk_, Aug 09 2006 *)

%Y Cf. A005478, A001605, A000045.

%K nonn

%O 1,1

%A _Lior Manor_, May 12 2004

%E More terms from _Alexander Adamchuk_, Aug 09 2006