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Fibonacci Prime — Definition, Formula & Examples

A Fibonacci prime is a number that belongs to the Fibonacci sequence and is also a prime number. For example, 2, 3, 5, 13, and 89 are all Fibonacci primes.

A Fibonacci prime is an element FnF_n of the Fibonacci sequence, defined by F1=1F_1 = 1, F2=1F_2 = 1, and Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} for n≥3n \geq 3, such that FnF_n is a prime number (having exactly two distinct positive divisors: 1 and itself).

Key Formula

Fn=Fn−1+Fn−2,F1=1,  F2=1F_n = F_{n-1} + F_{n-2}, \quad F_1 = 1,\; F_2 = 1
Where:
  • FnF_n = The nth Fibonacci number
  • nn = The position in the sequence (positive integer)

How It Works

To determine whether a Fibonacci number is a Fibonacci prime, you first generate the Fibonacci sequence and then test each term for primality. The early Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... From this list, you check which entries are prime. A useful fact: if nn is composite (and n>4n > 4), then FnF_n is also composite, so you only need to check Fibonacci numbers at prime indices (though a prime index does not guarantee a Fibonacci prime). It remains an open question whether infinitely many Fibonacci primes exist.

Worked Example

Problem: Is the 11th Fibonacci number a Fibonacci prime?
Step 1: Generate the Fibonacci sequence up to the 11th term.
1,1,2,3,5,8,13,21,34,55,891, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
Step 2: Identify the 11th term.
F11=89F_{11} = 89
Step 3: Test whether 89 is prime. Check divisibility by primes up to 89≈9.4\sqrt{89} \approx 9.4: 89 is not divisible by 2, 3, 5, or 7.
Answer: F11=89F_{11} = 89 is prime, so 89 is a Fibonacci prime.

Why It Matters

Fibonacci primes sit at the intersection of two fundamental structures in number theory — recursive sequences and prime numbers. They appear in competition math problems that ask students to connect divisibility properties of the Fibonacci sequence with primality testing. The open question of whether infinitely many exist illustrates how accessible-sounding problems can remain unsolved.

Common Mistakes

Mistake: Assuming every Fibonacci number at a prime index is itself prime.
Correction: A prime index is necessary (for FnF_n to be prime with n>4n > 4), but not sufficient. For example, F19=4181=37×113F_{19} = 4181 = 37 \times 113, which is composite despite 19 being prime.