Problem Statement

If we form a spiral with the integers, we find that prime numbers tend to fall on the diagonals of this spiral with a higher density than elsewhere in the spiral. This question asks to find how long it takes before the percentage of integers on both diagonals that are primes falls below 10%.

Solution Technique

Finding the numbers that lie on the diagonals of the spiral consists of iterating through a vector of integers at intervals of increasing spacing. So all we need to do is implement a counter to generate integers, a variable to keep track of how many integers until the next number on the diagonal, and a method to check if a number is prime.

Code

Here is the core of the method, which uses a queue to pop two numbers (east/west) at a time:

while(!q.isEmpty()) {

// Remove two diagonals (one row) at a time
// (problem asks for ratio of primes on *both* diagonals)
ned = q.remove(); numbers++;
nwd = q.remove(); numbers++;
row++;

if(Sieve.isPrime(ned)) {
primes++;
}
if(Sieve.isPrime(nwd)) {
primes++;
}

// Update ratio
ratio = (1.0*primes)/numbers;

if(row%1000==0) {
System.out.println(row);
}

// If < 10%, return
if(ratio <= 0.10) {
return Integer.toString(row+1);
}

// If not < 10%, increment i and compute next entries