Project Euler: Find 10.001st prime

Question

I have just started learning C# and decided to do some problems from Project Euler. I wrote a code to find 10 001st prime and I thought it would be cool optimize it to make it as fast as possible. How could I improve this to make it even faster?

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Diagnostics;

namespace primes
{
    class Program
    {
        static void Main(string[] args)
        {
            long prime = 2;
            long largestprime = 2;
            long potentialprime = 3;
            Console.WriteLine("Enter the prime you want to find: ");
            long primenum = Convert.ToInt64(Console.ReadLine());
            long[] primearray = new long[primenum + 1];
            primearray[1] = 2;
            Stopwatch sw = new Stopwatch();
            sw.Start();
            while (prime <= primenum)
            {
                bool isprime = true;
                for (long x = 1; x < prime; x += 3)
                {
                    if (primearray[x] * primearray[x] > potentialprime)
                         break;
                    if (prime > 3 && (potentialprime % primearray[x] == 0 || potentialprime % primearray[x+1] == 0 || potentialprime % primearray[x+2] == 0))
                         isprime = false;
                    if (prime <= 3 && potentialprime % primearray[x] == 0)
                         isprime = false;

                 }
                 if (isprime)
                {
                    primearray[prime] = potentialprime;
                    prime += 1;
                    largestprime = potentialprime;
                }
                potentialprime += 2;
            }
            sw.Stop();
            Console.WriteLine(sw.ElapsedMilliseconds + "ms to find ");
            Console.WriteLine(largestprime);
            Console.ReadKey();
         }
    }
}

Answer 1

As mentioned in the comments, you should go for more readable code:

1) Split by responsibility:

One method for user input:

ulong GetPrimeIndex() { // TODO: Get User input from Console }

The algorithm itself:

ulong FindNthPrime(ulong n) { // TODO: the algorithm }

print the result...

public static void Main(string[] args)
{ 
  ulong primeIndex = GetPrimeIndex();

  Stopwatch watch = Stopwatch.StartNew();
  ulong nthPrime = FindNthPrime(primeIndex);
  watch.Stop();

  Console.WriteLine($"{primeIndex} prime is: {nthPrime}");
  Console.WriteLine($"Elapsed time: {watch.ElapsedMilliseconds}");
}

2) Naming:

In C# local variable names use camelCase: largestprime should be largestPrime

---

Because primes are always positive, you should use ulong instead of long. It will give you a broader range to operate in.

---

I think you're missing a couple of breaks in these tests:

                if (primearray[x] * primearray[x] > potentialprime)
                     break;
                if (prime > 3 && (potentialprime % primearray[x] == 0 || potentialprime % primearray[x+1] == 0 || potentialprime % primearray[x+2] == 0))
                     isprime = false;
                if (prime <= 3 && potentialprime % primearray[x] == 0)
                     isprime = false;

It should be:

  if (primearray[x] * primearray[x] > potentialprime)
    break;
  if (prime > 3 && (potentialprime % primearray[x] == 0 || potentialprime % primearray[x + 1] == 0 || potentialprime % primearray[x + 2] == 0))
  {
    isprime = false;
    break;
  }
  if (prime <= 3 && potentialprime % primearray[x] == 0)
  {
    isprime = false;
    break;
  }

---

Be aware that

primearray[x] * primearray[x]

may overflow the domain range. You should maybe make the check by sqrt instead?