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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.

%I A245236 #17 Jul 17 2014 22:05:40
%S A245236 4,5,6,9,12,15,45,432,570
%N A245236 Numbers n such that the Fibonacci number F(n) satisfies F(n)^2+1 = f1*f2 where f1, f2 are prime Fibonacci numbers.
%C A245236 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 A245236 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 A245236 The corresponding Fibonacci numbers are 3, 5, 8, 34, 144, 610, 1134903170,...
%C A245236 If a(10) exists, it is greater than 30000. - _Robert Israel_, Jul 14 2014
%e A245236 4 is a term because F(4)^2+1 = F(3)*F(5)=> 3^2+1 = 2*5;
%e A245236 5 is a term because F(5)^2+1 = F(3)*F(7)=> 5^2+1 = 2*13;
%e A245236 6 is a term because F(6)^2+1 = F(5)*F(7)=> 8^2+1 = 5*13;
%e A245236 9 is a term because F(9)^2+1 = F(7)*F(11)=> 34^2+1 = 13*89;
%e A245236 12 is a term because F(12)^2+1 = F(11)*F(13)=> 144^2+1 = 89*233;
%e A245236 15 is a term because F(13)*F(17)=> 610^2+1 = 233* 1597.
%p A245236 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 A245236 # Alternative:
%p A245236 filter:= proc(n) uses combinat;
%p A245236     if n::even then isprime(n-1) and isprime(n+1) and isprime(fibonacci(n-1)) and isprime(fibonacci(n+1))
%p A245236   else isprime(n-2) and isprime(n+2) and isprime(fibonacci(n-2)) and isprime(fibonacci(n+2))
%p A245236 fi end proc:
%p A245236 select(filter, [$1..10^4]); # _Robert Israel_, Jul 14 2014
%Y A245236 Cf. A000045, A005478, A245306.
%K A245236 nonn,hard
%O A245236 1,1
%A A245236 _Michel Lagneau_, Jul 14 2014