大量循环耗时太长

Question

I'm working on a program designed to generate prime numbers, and I'm trying to find the 10001st prime number. Each time the loop runs, it tests variable i to see if it's divisible by variable j, which is part of another loop that goes up to half of i. If i is divisible by j then it adds one to variable prime, and when the loop's done it checks to see if variable prime is larger than two; if it is, then i is not prime. If it is prime, it gets push()ed to the variable array, which stores all the primes. Sorry if that's a little confusing.

My problem is that although this code works, whenever I set the first for loop too high (here I have it end at 100000), my browser freezes:

var prime = 0;
var array = [];
for(var i = 2; i <= 100000; i+=1) {
  var prime = 0;
  for(var j = 0; j <= i/2; j+=1) {
    if(i % j === 0) {
      prime += 1;
    }
  }

  if(prime < 2) {
    array.push(i);
  }
}

console.log(array.length)

I know I could just copy and paste all the prime numbers but I want to make the program work. Any ideas to make the program stop freezing?

Answer 1

This is because the time complexity of your algorithm is pretty much O(n^2). Sieve of Eratosthenes created a better algorithm that will prevent your browser from freezing. Here is one way of doing it:

var exceptions = {
  2: 2,
  3: 3,
  5: 5,
  7: 7
}

var primeSieve = function (start, end) {
  var result = [];
  for (start; start < end; start++) {
    if (start in exceptions) result.push(start);
    if (start > 1 && start % 2 !== 0 && start % 3 !== 0 && start % 5 !== 0 && start % 7 !== 0) {
      result.push(start);
    }
  }
  return result;
};

You can obviously make this look prettier, but I just did it quickly so you can get the point. I hope this helps! Let me know if you have further questions.