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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

%I

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

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

%C 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.

%C 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).

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

%C If a(10) exists, it is greater than 30000. - _Robert Israel_, Jul 14 2014

%e 4 is a term because F(4)^2+1 = F(3)*F(5)=> 3^2+1 = 2*5;

%e 5 is a term because F(5)^2+1 = F(3)*F(7)=> 5^2+1 = 2*13;

%e 6 is a term because F(6)^2+1 = F(5)*F(7)=> 8^2+1 = 5*13;

%e 9 is a term because F(9)^2+1 = F(7)*F(11)=> 34^2+1 = 13*89;

%e 12 is a term because F(12)^2+1 = F(11)*F(13)=> 144^2+1 = 89*233;

%e 15 is a term because F(13)*F(17)=> 610^2+1 = 233* 1597.

%p 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:

%p # Alternative:

%p filter:= proc(n) uses combinat;

%p if n::even then isprime(n-1) and isprime(n+1) and isprime(fibonacci(n-1)) and isprime(fibonacci(n+1))

%p else isprime(n-2) and isprime(n+2) and isprime(fibonacci(n-2)) and isprime(fibonacci(n+2))

%p fi end proc:

%p select(filter, [$1..10^4]); # _Robert Israel_, Jul 14 2014

%Y Cf. A000045, A005478, A245306.

%K nonn,hard

%O 1,1

%A _Michel Lagneau_, Jul 14 2014

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Last modified February 26 03:09 EST 2020. Contains 332272 sequences. (Running on oeis4.)