Prime Numbers

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Positive integers greater than 1 that have no divisors other than 1 and themselves.

A prime number is a positive integer p > 1 whose only positive divisors are 1 and p itself. The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Note that 2 is the only even prime; every other prime is odd. Primes are not generated by a simple closed-form formula — you must check divisibility (or consult a known list) to determine the next term.

p ∈ ℕ, p > 1, divisors(p) = {1, p}

• 2, 3, 5, 7, 11 → next is 13 (skipping 12 = 2 · 6)
• 11, 13, 17, 19, 23 → next is 29 (skipping 24, 25, 26, 27, 28)
• 5, 7, 11, 13, 17 → gaps of 2, 4, 2, 4 (prime gaps)

How to recognize it

• Memorize the first primes up to 100 (only 25 of them)
• A number n > 1 is prime if no integer from 2 to √n divides it
• All primes except 2 and 3 have the form 6k ± 1 (but not every 6k ± 1 is prime)

Common mistakes

• Thinking 1 is prime — it is not; primes must be > 1
• Thinking 9, 15, 21, 25, 27, 33, 35 are prime (all composite)
• Assuming all odd numbers are prime (9 = 3 × 3 is the first counterexample)

Practice guidance

Number Sequences