From the people behind the Alan Turing Cryptography Competition.
You are reading the website of the 2020 edition of the MathsBombe Competition, which ended on Sunday 26 April at 11:59 pm

# Puzzle 8

The Fibonacci sequence of numbers is well-known: $$F_1=1,F_2=1,F_3=2,F_4=3,F_5=5,F_6=8,F_7=13,….$$

The nth term, $$F_n$$, in this sequence is given by the recurrence relation $$F_n=F_{n-1}+F_{n-2}$$ with initial terms $$F_1=1,F_2=1$$. However, there is nothing special about starting with 1 and 1; you could start with any two numbers for F_1,F_2, although this will, of course, give you a different sequence.

Suppose you pick two positive integers p,q and define the sequence $$F_1=p,F_2=q,F_n=F_{n-1}+F_{n-2}.$$

Suppose this sequence has the property that $$F_m=1000000$$ for some value of m. What are the values of p,q that make m as large as possible?

You should enter your answer in the form p,q (two integers separated by a comma, with no spaces). For example, if you think the answer is $$p=12, q=34$$ then you should enter 12,34.