

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




OFFSET

1,1


COMMENTS

Or index i of any Fibonacci number F(i) such that F(i1) and F(i+1) are primes if i is even or F(i2) and F(i+2) are primes if i is odd where F(i) is the ith Fibonacci number.
In the general case, F(i+1)*F(i1) = F(i)^2 + 1 if i even or F(i+2)*F(i2) = 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


LINKS

Table of n, a(n) for n=1..9.


EXAMPLE

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.


MAPLE

with(combinat, fibonacci):with(numtheory):nn:=1000:for n from 1 to nn do:if (type(fibonacci(n+1), prime) and type(fibonacci(n1), prime) and irem(n, 2)=0) or (type(fibonacci(n+2), prime) and type(fibonacci(n2), prime) and irem(n, 2)=1) then print(n):else fi:od:
# Alternative:
filter:= proc(n) uses combinat;
if n::even then isprime(n1) and isprime(n+1) and isprime(fibonacci(n1)) and isprime(fibonacci(n+1))
else isprime(n2) and isprime(n+2) and isprime(fibonacci(n2)) and isprime(fibonacci(n+2))
fi end proc:
select(filter, [$1..10^4]); # Robert Israel, Jul 14 2014


CROSSREFS

Cf. A000045, A005478, A245306.
Sequence in context: A192360 A115984 A145025 * A209122 A073263 A039013
Adjacent sequences: A245233 A245234 A245235 * A245237 A245238 A245239


KEYWORD

nonn,hard


AUTHOR

Michel Lagneau, Jul 14 2014


STATUS

approved



