Odd Prime — Definition, Formula & Examples
An odd prime is any prime number that is not 2. Since 2 is the only even prime, every other prime — 3, 5, 7, 11, 13, and so on — is called an odd prime.
An odd prime is an integer such that has exactly two positive divisors: 1 and itself. Equivalently, it is a prime number that is not divisible by 2.
How It Works
To determine whether a number is an odd prime, first check that it is odd (not divisible by 2), then confirm it is prime (divisible only by 1 and itself). The number 2 is prime but even, so it does not qualify. Mathematicians use the phrase "odd prime" frequently because many theorems in number theory work differently for 2 than for other primes, so it is convenient to have a short name for all primes except 2.
Worked Example
Problem: Is 17 an odd prime?
Check if odd: 17 divided by 2 gives a remainder of 1, so 17 is odd.
Check if prime: Test divisibility by all primes up to the square root of 17 (which is about 4.1). 17 is not divisible by 2 or 3.
Conclude: Since 17 is both odd and prime, it is an odd prime.
Answer: Yes, 17 is an odd prime.
Why It Matters
Many results in number theory only apply to odd primes. For example, every odd prime can be written as either or , a classification that drives important theorems about which numbers are sums of two squares. Recognizing the special role of 2 helps you apply these results correctly.
Common Mistakes
Mistake: Thinking 2 is an odd prime because it is prime.
Correction: 2 is the only even prime. An odd prime must be both prime and odd, so 2 is excluded by definition.