A245236 Numbers n such that the Fibonacci number F(n) satisfies F(n)^2+1 = f1*f2 where f1, f2 are prime Fibonacci numbers.

4, 5, 6, 9, 12, 15, 45, 432, 570

Offset: 1

Author

Michel Lagneau, Jul 14 2014

Keywords

• nonn
• hard

Comments

Or index i of any Fibonacci number F(i) such that F(i-1) and F(i+1) are primes if i is even or F(i-2) and F(i+2) are primes if i is odd where F(i) is the i-th Fibonacci number.

In the general case, F(i+1)*F(i-1) = F(i)^2 + 1 if i even or F(i+2)*F(i-2) = F(i)^2 + 1 if i odd (Cassini’s identity).

The corresponding Fibonacci numbers are 3, 5, 8, 34, 144, 610, 1134903170,...

If a(10) exists, it is greater than 30000. - Robert Israel, Jul 14 2014

Examples

			4 is a term because F(4)^2+1 = F(3)*F(5)=> 3^2+1 = 2*5;
5 is a term because F(5)^2+1 = F(3)*F(7)=> 5^2+1 = 2*13;
6 is a term because F(6)^2+1 = F(5)*F(7)=> 8^2+1 = 5*13;
9 is a term because F(9)^2+1 = F(7)*F(11)=> 34^2+1 = 13*89;
12 is a term because F(12)^2+1 = F(11)*F(13)=> 144^2+1 = 89*233;
15 is a term because F(13)*F(17)=> 610^2+1 = 233* 1597.
		

Crossrefs

Cf. A000045, A005478, A245306.

Programs

Maple

with(combinat,fibonacci):with(numtheory):nn:=1000:for n from 1 to nn do:if (type(fibonacci(n+1),prime) and type(fibonacci(n-1),prime) and irem(n,2)=0) or (type(fibonacci(n+2),prime) and type(fibonacci(n-2),prime) and irem(n,2)=1) then print(n):else fi:od:
# Alternative:
filter:= proc(n) uses combinat;
    if n::even then isprime(n-1) and isprime(n+1) and isprime(fibonacci(n-1)) and isprime(fibonacci(n+1))
  else isprime(n-2) and isprime(n+2) and isprime(fibonacci(n-2)) and isprime(fibonacci(n+2))
fi end proc:
select(filter, [$1..10^4]); # Robert Israel, Jul 14 2014