How to Check If a Number Is Prime: Efficient Methods & Code

Picture this: I'm sitting in my first coding interview years ago, sweating bullets when they ask "How would you check if 1,000,000 is prime?" I mumbled something about dividing by every number until my brain hurt. Later that week, my math professor saw me brute-forcing primes and gave me that disappointed head-shake. That's when I realized most tutorials skip the practical reality of prime checking.

Let's fix that. Whether you're a programmer optimizing algorithms, a student tackling number theory, or just curious, we'll cut through the fluff. I'll share the methods that actually work in practice – including where textbooks get it wrong and how to avoid common pitfalls.

What Exactly Makes a Number Prime?

Prime numbers are stubborn. They refuse to play nice with division. Formally: a prime number has exactly two distinct divisors – 1 and itself. But let's break that down:

I used to think primes were rare unicorns until I calculated that about 25% of numbers under 100 are prime. Still surprises people when I mention it!

Fun fact: 2 is the weirdo – the only even prime. All other evens are divisible by 2, so they're automatically out.

Why Bother Checking Primes Anyway?

"When will I ever need this?" I groaned in 10th grade math class. Turns out:

Last month, my friend wasted 3 hours debugging because his code treated 1 as prime. Don't be that person.

Trial Division: The Hands-On Method

Here's where many tutorials go wrong. They describe trial division like this:

"Divide n by all integers from 2 to n-1."

That's inefficient and painful. Let me show you the smarter way.

Step-by-Step Process for Humans

Check if the number is:

• Less than 2 → Not prime
• Even (except 2) → Not prime
• Ends with 5 (except 5) → Not prime

Only test divisors from 3 to the square root of n, skipping even numbers. Why sqrt(n)? Because if n has a factor larger than sqrt(n), it must have a corresponding factor smaller than sqrt(n).

When Trial Division Works Best

• Numbers under 10 million
• Quick mental checks (for small numbers)
• Learning the fundamentals

I use this method daily for quick checks. Yesterday I verified 127 was prime while waiting for coffee – took 15 seconds.

Code Implementation: From Pseudocode to Real Languages

Here's how to translate this logic into code. I'll show Python because it reads like plain English, but the logic transfers anywhere.

def is_prime(n):
    if n < 2:
        return False
    if n in (2, 3):
        return True
    if n % 2 == 0:
        return False
    # Check odd divisors up to sqrt(n)
    i = 3
    while i * i <= n:
        if n % i == 0:
            return False
        i += 2
    return True

Execution Time Comparison:

JavaScript version for web developers:

function isPrime(num) {
  if (num <= 1) return false;
  if (num === 2) return true;
  if (num % 2 === 0) return false;
  const limit = Math.sqrt(num);
  for (let i = 3; i <= limit; i += 2) {
    if (num % i === 0) return false;
  }
  return true;
}

Advanced Methods for Big Numbers

When numbers get huge (like 60+ digits), trial division chokes. That's when you need heavier machinery.

Probabilistic Tests: Miller-Rabin

This method gives a "probably prime" result with adjustable certainty. Used in cryptography libraries like OpenSSL.

How it works:

1. Write n-1 as (2^s)*d
2. Pick random witness a between 2 and n-2
3. Compute a^d mod n
4. If result isn't 1 or n-1, n is composite

Honestly? Unless you're working with 256-bit primes, this is overkill. I've only needed it twice in 10 years of coding.

Sieve Methods

Need all primes up to N? Sieves are your friend. The Sieve of Eratosthenes is surprisingly efficient:

Python sieve implementation:

def primes_up_to(n):
    sieve = [True] * (n+1)
    sieve[0] = sieve[1] = False
    for i in range(2, int(n**0.5)+1):
        if sieve[i]:
            sieve[i*i:n+1:i] = [False]*len(sieve[i*i:n+1:i])
    return [i for i, prime in enumerate(sieve) if prime]

Special Cases That Trip Everyone Up

These numbers break naive implementations. Bookmark this section.

Frequently Asked Questions (with Real Answers)

Performance Considerations

Different methods for different needs:

The AKS test is theoretically beautiful but painfully slow in practice. I once ran it on a 30-digit number – took 12 minutes versus Miller-Rabin's 0.2 seconds.

Closing Thoughts from Experience

After years of implementing primality tests, here's my cheat sheet:

• For daily use: optimized trial division covers 99% of cases
• Remember: Skip even numbers after 2
• Stop at sqrt(n) – seriously, just do it
• Test edge cases: 0, 1, 2, negatives
• When in doubt, test 91 (7×13) – catches many broken implementations

The most satisfying moment? When I helped a student discover that 1234567891 is prime using these methods. Her excitement reminded me why primes fascinate us – they're nature's uncrackable codes.

Got a tricky number? Plug it into this simple Python checker. Better yet, implement your own. Nothing solidifies understanding like making the mistakes yourself.