# Project Euler 7 Solution: 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?

**Solution**

The easiest way to solve this problem is checking number by number if it's a prime and if so, incrementing a counter until 10001. Since every prime after 2 is odd, we can increment by two, which halves the actual search space. However, it is also known that all primes except 2 and 3 have the form \(6k\pm 1\), which allows us to go in steps of 6. An implementation can then look as follows:

function solution(L) { var c = 2; var n = 0; while (c < L) { n+= 6; if (isPrime(n + 1)) { c++; } if (isPrime(n - 1)) { c++; } } // Add one for the final prime being of the form 6k + 1 return n + 1; } solution(10001);

A function isPrime is used in the solution. A primal check can be done by looping from 2 to \(n\) and check if any number on the way divides our number. If not, we found a prime. One optimization is to loop to \(\sqrt{n}\) instead of the whole space, since only multiples of already known primes remain above the limit. What we also can do is unrolling checks of multiples of 2 and 3, which allows us to loop in a stepwidth of 6, which however requires a check of every \(i+2\) as well. The implementation can then be stated as:

function isPrime(n) { if (n < 2) return false; if (n % 2 === 0) return n === 2; if (n % 3 === 0) return n === 3; var h = Math.floor(1 + Math.sqrt(n)); var i = 5; while (i <= h) { if (n % i === 0) return false; if (n % (i + 2) === 0) return false; i+= 6; } return true; }

As such, the complexity of isPrime is bound to \(O(\sqrt{n})\), which results in an overall complexity of \(O(n\sqrt{n})\). Using a sieve could speed up things here, but utilizes a space complexity of \(O(n)\).