Ian Dorian Macleod
Problem 25
Problem Statement

The Fibonacci sequence is defined by the recurrence relation:

Fn = Fn−1 + Fn−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 index of the first term in the Fibonacci sequence to contain 1000 digits?

url: https://projecteuler.net/problem=25
Approach

C++ implementation

#include <stdc++.h> 
using namespace std;

// note: this can also be represented using a convergence theorem
// on the reccurence relation where F_n = (\phi_n / \sqrt(5))

// assume a >= b 
vector add(vector a, vector b) {
    vectorans;
    int carry = 0;
    for(int i = 0; i < a.size(); i++) {
        if (i > b.size()-1) {
            int tmp = (a[i] + carry)%10;
            ans.push_back(tmp);
            carry = (a[i] + carry) / 10;
        }
        else {
            int tmp = (a[i] + b[i] + carry) % 10;
            ans.push_back(tmp);
            carry = (a[i] + b[i] + carry) / 10;
        }
    }
    if (carry) ans.push_back(carry);
    return ans;
}

int fib_digits() {
    vectorlast_last = {1};
    vectorlast = {1};
    int i = 2;
    while(last.size() < 1000) {
        vectortmp = add(last, last_last);
        last_last = last;
        last = tmp;
        i++;
    }
    return i;
}

int main() {
    cout << fib_digits() << endl;
    return 0;
}

You can also download the source code for this problem here and compile it on your local machine.
Further Analysis