A371866 Primes of the form Fibonacci(m^(k+1))/Fibonacci(m^k), where m > 1 and k >= 1.

3, 7, 17, 47, 2207, 97415813466381445596089

Offset: 1

Author

Robert Israel, Apr 09 2024

Keywords

• nonn

Comments

a(7) > 10^25000 if it exists.

m must be prime, as Fibonacci((a*b)^(k+1))/Fibonacci((a*b)^k) = (Fibonacci((a*b)^(k+1))/Fibonacci(a^k * b^(k+1))) * Fibonacci(a^k * b^(k+1))/Fibonacci((a*b)^k).

Examples

			a(1) = 3 = F(2^2)/F(2^1) where F = Fibonacci.
a(2) = 7 = F(2^3)/F(2^2).
a(3) = 17 = F(3^2)/F(3^1).
a(4) = 47 = F(2^4)/F(2^3).
a(5) = 2207 = F(2^5)/F(2^4).
a(6) = 97415813466381445596089 = F(11^2)/F(11^1).
		

Crossrefs

Primes in A181419.

Cf. A000045.

Programs

Maple

N:= 10^1000: # for terms < N
R:= NULL: F:= combinat:-fibonacci:
p:= 1:
do
 p:= nextprime(p);
 v:= F(p);
 for k from 2 do
   w:= v;
   v:= F(p^k);
   r:= v/w;
   if r > N then break fi;
   if isprime(r) then R:= R, r fi;
 od;
 if k = 2 then break fi;
od:
sort([R]);