Yes, by definition, a prime number must be an integer.
Here's a breakdown:
-
Definition of a Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
-
Natural Numbers are Integers: Natural numbers (1, 2, 3, ...) are a subset of integers. Integers include all positive and negative whole numbers and zero (... -3, -2, -1, 0, 1, 2, 3, ...).
-
Implication: Since a prime number is a natural number, and natural numbers are integers, a prime number is therefore an integer. Prime numbers cannot be fractions, decimals, or irrational numbers.
For Example:
- 2, 3, 5, 7, 11, 13 are all integers and prime numbers.
- 2.5 is not an integer and therefore cannot be a prime number.
- 1/2 is not an integer and therefore cannot be a prime number.
- √2 is not an integer and therefore cannot be a prime number.
In summary, the foundational definition of prime numbers inherently requires them to be integers (specifically, natural numbers greater than 1).
