Fibonacci Checker

What is the Fibonacci Sequence?

The Fibonacci sequence is one of the most famous number sequences in mathematics. Each number is the sum of the two preceding ones, starting from 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

Named after Leonardo of Pisa (known as Fibonacci), who introduced it to Western mathematics in his 1202 book Liber Abaci, this sequence appears throughout nature, art, and architecture.

Fibonacci in Geocaching

Fibonacci numbers frequently appear in geocaching puzzles:

• Direct clues: Coordinates based on Fibonacci numbers
• Pattern puzzles: Continuing a Fibonacci-like sequence
• Golden ratio: Related calculations using φ ≈ 1.618
• Nature themes: Puzzles about spirals, petals, seeds
• Position puzzles: Using F(n) to encode values

How to Check if a Number is Fibonacci

A positive integer n is a Fibonacci number if and only if one of these is a perfect square:

• 5n² + 4
• 5n² − 4

This mathematical property allows quick checking without generating the entire sequence.

The Golden Ratio Connection

The ratio of consecutive Fibonacci numbers approaches the Golden Ratio (φ):

φ = (1 + √5) / 2 ≈ 1.6180339887...

For large n: F(n+1) / F(n) ≈ φ

This ratio appears in nature, art, and architecture—from nautilus shells to the Parthenon.

Fibonacci Properties

• Every 3rd Fibonacci is divisible by 2
• Every 4th Fibonacci is divisible by 3
• Every 5th Fibonacci is divisible by 5
• GCD(F(m), F(n)) = F(GCD(m, n))
• Sum of first n Fibonacci = F(n+2) - 1

Fibonacci in Nature

Fibonacci numbers appear in surprising natural phenomena:

• Flower petals: Lilies (3), buttercups (5), delphiniums (8), marigolds (13)
• Seed heads: Sunflower spirals often have 34/55 or 55/89 spirals
• Pinecones: Spiral counts typically follow Fibonacci
• Tree branches: Growth patterns often follow the sequence
• Shells: Nautilus shells approximate golden spirals

Extended Fibonacci Sequences

Variations of the Fibonacci sequence include:

• Lucas Numbers: Start with 2, 1 instead of 0, 1
• Tribonacci: Sum of previous 3 numbers
• Negafibonacci: Extended into negative indices
• Generalized Fibonacci: Any starting values

Binet's Formula

The nth Fibonacci number can be calculated directly using Binet's formula:

F(n) = (φⁿ - ψⁿ) / √5

Where φ = (1+√5)/2 and ψ = (1-√5)/2

Related Mathematical Tools

• Prime Checker: Many Fibonacci numbers are also prime (Fibonacci primes).
• Digital Root: Fibonacci numbers have interesting digital root patterns.
• GCD/LCM Calculator: Fibonacci numbers have special GCD properties.