feat(problem): 7 - 10,001st prime

src/bin/7.rs

@@ -0,0 +1,64 @@
+/*
+Problem 7 - 10001st prime
+
+By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
+What is the 10 001st prime number?
+*/
+
+use std::f64::consts::E;
+
+// Implementation of the Sieve of Eratosthenes
+// https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
+fn find_primes(upper_bound: usize) -> Vec<usize> {
+	let mut values: Vec<bool> = (2..(upper_bound + 1)).map(|_| true).collect();
+	let mut i = 2;
+
+	// Adjust values array
+	while i < (upper_bound as f64).sqrt() as usize {
+		if values[i] {
+			let mut j = i.pow(2);
+
+			while j < upper_bound {
+				values[j] = false;
+				j += i;
+			}
+		}
+
+		i += 1;
+	}
+
+	// Find the indexes where the values are true
+	let mut primes: Vec<usize> = vec![];
+
+	for i in 0..values.len() {
+		if values[i] {
+			primes.push(i);
+		}
+	}
+	
+	return primes;
+}
+
+// Bounds for nth prime: http://en.wikipedia.org/wiki/Prime_number_theorem
+// ln n + ln(ln n) - 1 < p_n / n < ln n + ln(ln n), n >= 6
+fn upper_bound_for_nth_prime(n: usize) -> usize {
+	// The first 5 primes are under 13 (6th prime) 
+	if n < 6 {
+		return 13;
+	}
+
+	let ln_n = (n as f64).log(E);
+
+	return n * (ln_n + ln_n.log(E)).ceil() as usize;
+}
+
+fn nth_prime(n: usize) -> usize {
+	let primes = find_primes(upper_bound_for_nth_prime(n));
+	return primes[n - 1];
+}
+
+fn main() {
+	let number = nth_prime(10001);
+
+	println!("The 10,001st prime number is {number}");
+}