Prime Numbers Formula

The Formula

p is prime if p > 1 and its only positive divisors are 1 and p

When to use: Primes can't be broken down further—they're the 'atoms' of multiplication.

Quick Example

2, 3, 5, 7, 11, 13, 17, 19, 23... (2 is the only even prime)

Notation

p typically denotes a prime; primality is tested by checking divisors up to \sqrt{p}

What This Formula Means

Integers greater than 1 whose only positive divisors are 1 and themselves—they cannot be factored further.

Primes can't be broken down further—they're the 'atoms' of multiplication.

Formal View

p \in \mathbb{Z} is prime \iff p > 1 and \forall\, a, b \in \mathbb{Z}^+,\; p = ab \implies a = 1 \text{ or } b = 1. Fundamental Theorem of Arithmetic: every n > 1 factors uniquely as n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}.

Worked Examples

Example 1

easy
Find the prime factorization of 180.

Solution

  1. 1
    Divide by 2: 180 \div 2 = 90. Divide again: 90 \div 2 = 45.
  2. 2
    45 is odd, try 3: 45 \div 3 = 15. Again: 15 \div 3 = 5.
  3. 3
    5 is prime, so stop. Prime factorization: 180 = 2^2 \times 3^2 \times 5.

Answer

2^2 \times 3^2 \times 5
To find the prime factorization, repeatedly divide by the smallest prime that divides evenly, working upward. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).

Example 2

medium
Determine whether 97 is prime.

Common Mistakes

  • Thinking 1 is a prime number — by definition, primes must be greater than 1 (1 has only one factor, not exactly two)
  • Believing all odd numbers are prime — 9 is odd but not prime (9 = 3 \times 3), and 15 is odd but not prime (15 = 3 \times 5)
  • Saying 2 is not prime because it is even — 2 is the only even prime number and is the smallest prime

Why This Formula Matters

Every number factors uniquely into primes (Fundamental Theorem of Arithmetic); primes are the basis of modern cryptography.

Frequently Asked Questions

What is the Prime Numbers formula?

Integers greater than 1 whose only positive divisors are 1 and themselves—they cannot be factored further.

How do you use the Prime Numbers formula?

Primes can't be broken down further—they're the 'atoms' of multiplication.

What do the symbols mean in the Prime Numbers formula?

p typically denotes a prime; primality is tested by checking divisors up to \sqrt{p}

Why is the Prime Numbers formula important in Math?

Every number factors uniquely into primes (Fundamental Theorem of Arithmetic); primes are the basis of modern cryptography.

What do students get wrong about Prime Numbers?

1 is NOT prime—primes need exactly two distinct factors. And 2 is the only even prime; every other even number has 2 as a factor.

What should I learn before the Prime Numbers formula?

Before studying the Prime Numbers formula, you should understand: factors, divisibility intuition.

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