# Integer Factorization using the Trial Division Algorithm

I am trying to factor integers using the trial division algorithm (Wikipedia: http://en.wikipedia.org/wiki/Trial_division).

I reviewed the Wikipedia entry for trial division and multiple other sources but I don't completely understand the trial division algorithm.

I wrote the code below based on my findings and it finds all of the prime factors of a number.

For example, input 20 and you get its prime factors 2 2 5. I would like to know if it can be simplified or better written (I want to stick with the trial division algorithm --- I don't plan on inputting numbers any larger than 9,999)?

    <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<title>Prime Number Checker</title>
<script>
// All the prime numbers under 1000
var primeNumbers = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997];

// Find prime factors
var n = 20;

var primeFactors = "";

// Trial division algorithm
for (var i = 0; primeNumbers[i] < n; i++) {
if (n % primeNumbers[i] == 0) {
var temp = n;

while (temp % primeNumbers[i] == 0) {
primeFactors += " " + primeNumbers[i] + " ";
}
}
}

document.getElementById("divSolution").innerHTML = primeFactors;
}
</script>

<body>
<div id="divSolution"></div>
</body>
</html>


1. Before writing any code, perform the computation manually. How would you do it for 20? You'd do something along the lines like :

• let's try the first prime number : 2
• is 2 a divisor of 20 ? yes, it is. 20 is 2*10 so let's add 2 to the list of divisors and let's focus on 10 now.
• is 2 a divisor of 10 ? yes, it is. 10 is 2*5 so let's add 2 to the list of divisors and let's focus on 5 now.
• let's consider the next prime number. It's 3. Oh, 3*3 is actually bigger than 5 so we won't be able to find any other prime factor smaller than 5 : let's add 5 to the list of divisors.
• Try it yourself with numbers such a 8, 100, 19....
2. Then, about your code : the first is I noticed that your loop condition is not right and you might get out of your array. Just try a value of n greater than 168 and you'll probably notice the issue.

If you want to follow the wiki page and to something like : for p in primes: if p*p > n: break what you really need to do is.

// Find prime factors
var n = 180;
var primeFactors = "";
// Trial division algorithm
while (n % p == 0) {
primeFactors += " " + p + " ";
n/=p;
}
}
if (n>1)
{
primeFactors += " " + n + " ";
}

document.getElementById("divSolution").innerHTML = primeFactors;


(Sorry for rewriting the whole code)