Grok (XAI)

The fundamental relationship is expressed by the following theorem:

Theorem: If Fₙ is prime, then n is either prime or n = 4.

This arises from a key divisibility property of the Fibonacci sequence: if m divides n, then Fₘ divides Fₙ. When n is composite and n > 4, the smallest prime divisor p of n (p ≥ 3) satisfies Fₚ > 1, and Fₚ divides Fₙ with quotient greater than 1, making Fₙ composite.

Thus, Fibonacci primes can only occur at prime indices (with the single exception of n = 4).

Exceptions and Deviations

The only exception to the theorem is n = 4: 4 is composite, yet F₄ = 3 is prime. This occurs because the smallest prime divisor of 4 is 2, and F₂ = 1 (which does not force compositeness). No other composite index yields a prime Fibonacci number; this is mathematically proven.

Deviations in the opposite direction are common: many prime indices produce composite Fibonacci numbers. Examples include:

• n = 31 (prime) → F₃₁ = 1,346,269 = 557 × 2,417 (composite)
• n = 37 (prime) → F₃₇ = 24,157,817 = 73 × 331,067 (composite)
• n = 41 (prime) → F₄₁ = 267,914,296 (composite, with known factors)

Such cases increase with larger n.

Evaluation of Rarity

The extreme rarity of Fibonacci primes stems from several factors:

1. Exponential growth limits the sequence to only 45 terms up to one billion.
2. Primality is structurally constrained to prime indices (except n = 4).
3. Even at prime indices, large Fibonacci numbers typically possess algebraic factors or are composite by chance. All proven Fibonacci primes have indices ≤ 43; no larger proven examples exist.

These structural and probabilistic constraints make the intersection of the two sets exceptionally small.

This article was prepared by Grok, an artificial intelligence model developed by xAI. Completion date: 12.02.2026, time: 13:18. Operating mode: Standard performance level.

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