Project Euler 104: Find the first Fibonacci number for which the ends are pandigital.

Problem Description

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_n−1 + F_n−2, where F₁ = 1 and F₂ = 1.

It turns out that F₅₄₁, which contains 113 digits, is the first Fibonacci number for which the last nine digits are 1-9 pandigital (contain all the digits 1 to 9, but not necessarily in order). And F₂₇₄₉, which contains 575 digits, is the first Fibonacci number for which the first nine digits are 1-9 pandigital.

Given that F_k is the first Fibonacci number for which the first nine digits AND the last nine digits are 1-9 pandigital, find k.

Analysis

Even calculating the Fibonacci sequence without any optimization this solution was lightning fast. Getting the bottom 9 digits was easy and has been done in previous problems. The top 9 required the formula:

phi = (1 + sqrt(5)) / 2
t = n * log10(phi) + log10(1/sqrt(5))
t = int((pow(10, t – int(t) + 8)))

So, after finding a candidate with the bottom 9 digits pandigital we query the formula to see if the top 9 are pandigital. The first one found will be our first solution.

Project Euler 104 Solution

Afterthoughts

• The constants in this solution are from the formula: t = n * log10(phi) + log10(1/sqrt(5)).
  
• Function is_pandigital is listed in Common Functions and Routines for Project Euler