Numbers n such that Fib(2n-1) and Fib(2n+1) are both prime.

2, 3, 6, 216, 285

1

No other terms among known Fibonacci primes. These numbers are the solutions to sum_{d | n, d < n} d^2 = 3*n. Cai, Chen, and Zhang prove that all solutions to this equation are the product of two prime Fibonacci numbers Fib(2n-1) and Fib(2n+1). They prove that there are only a finite number of solutions.

T. D. Noe, Plot of 5 terms

T. Cai, D. Chen and Y. Zhang, Perfect numbers and Fibonacci primes (I), to appear in Int. J. Number Theory.

Tianxin Cai, Liuquan Wang, And Yong Zhang, Perfect numbers and Fibonacci primes (II), arXiv 1406.5684, Jun 22 2014.

(Mma) Select[Range[1000], PrimeQ[Fibonacci[2*#-1]] && PrimeQ[Fibonacci[2*#+1]] &]

Cf. A000045 (Fibonacci numbers), A001605 (indices of Fibonacci primes), S000102.

nonn,fini,full

T. D. Noe, Jun 23 2014