Project Euler

Problem 25: 1000-digit Fibonacci Number

The Fibonacci sequence is defined by the recurrence relation: F_n = F_{n - 1} + F_{n - 2}, where F₁ = 1 and F₂ = 1. Hence the first 12 terms will be:
F₁ = 1
F₂ = 1
F₃ = 2
F₄ = 3
F₅ = 5
F₆ = 8
F₇ = 13
F₈ = 21
F₉ = 34
F₁₀ = 55
F₁₁ = 89
F₁₂ = 144
The 12th term, F₁₂, is the first term to contain three digits. What is the first term in the Fibonacci sequence to contain 1000 digits?

The Catch

How to efficiently generate 1,000-digit Fibonacci sequence.

The Light

Of course we can use Java's BigInteger class and iteratively generate the Fibonacci sequence until the first instance of a 1,000-digit term appears. However, such an approach might take a few hundred milliseconds.

The article on Fibonacci Number from Wolfram Alpha offers the Binet's Fibonacci Formula, which offers a closed form formula for generating the Fibonacci sequence: Thus, finding n would give the index of the first Fibonacci number that has 1,000 digits.

The Code

public class Problem25
{
  public static void main(String[] args)
  {
    System.out.println(Math.log10(10) * 999.0 + Math.log10(5)/2)/Math.log10(1.61803398875);
  }
}