Hi guys,

Here's a problem from the 2016 BIO round 1.

Problem statement

A prime number is a whole number, greater than 1, that can only be divided by itself and the number 1. Two prime numbers are connected if the difference between them is 2n for some whole number n ≥ 0; e.g. possible differences are 1, 2, 4, 8, 16, 32, … A path is a sequence of (at least two) prime numbers, without repetition, where adjacent numbers in the sequence are connected. If the first number in the sequence is p and the last number is q then we say the path is between p and q. The length of a path is the total number of prime numbers used. There may be multiple paths between two prime numbers; the lengths of these paths may be different. For example:

• 13 is connected to 5 (13 — 5 = 8 = 23), 5 is connected to 3 (5 — 3 = 2 = 21) and 3 is connected to 2 (3 — 2 = 1 = 20);

• As 13 and 5 are connected there is a path between them (13—5) whose length is 2;

• There is a path from 13 to 2 (13—5—3—2) whose length is 4;

• There is a longer path from 13 to 2 (13—17—19—3—2) whose length is 5.

You will be given an upper limit on the primes you are allowed to use. For example, if the limit was 18 then the path 13—17—19—3—2 would not be permitted as it includes a prime above this limit.

Write a program to determine the length of the shortest path between two primes. Your program should input three integers in order: l (4 ≤ l ≤ $$$2^{24}$$$) indicating the highest value you are allowed to use, followed by the primes p then q (2 ≤ p < q < l). You will only be given input where there is a path between p and q using values below l. You should output the length of the shortest path.

My question

It should be an easy question, just breadth first search from p to q with a few tweaks.

My issue is the "(4 ≤ l ≤ $$$2^{24}$$$)". For my search to work it needs to be able to calculate which values are prime and which are not. My solution was to just use a sieve to pre-calculate them all. However, that obviously wont work for $$$l=2^{24}$$$ as the mark scheme only gives points to solutions that finish in under 1 second.

One thing to note is that the actual largest value in the test cases was l=1,000,000 so it could be that a solution for larger l doesn't exist. What do you think?