Is 2 prime?

This is part of a series on common misconceptions.

True or False?

The number $2$ is prime.

Why some people say it's true: It is so small, it's prime.

Why some people say it's false: Even numbers are not prime.

The statement is $\color{#20A900}{\textbf{true}}.$ (But it is not at all based on "being too small." See Graham’s number.)

Proof: The definition of a prime number is a positive integer that has exactly two distinct divisors. Since the divisors of 2 are 1 and 2, there are exactly two distinct divisors, so 2 is prime.

Rebuttal: Because even numbers are composite, 2 is not a prime.

Reply: That is true only for all even numbers greater than 2. If a number is of the form $n = 2k$ with $k > 1$, then we know it has the distinct factors 1, 2, and $2k$, and thus it cannot be prime. However, for $k =1$, we have $2 = 2k,$ so there are only 2 factors. In fact, the only reason why most even numbers are composite is that they are divisible by 2 (a prime) by definition. Are all multiples of 3 composite? No. Does three not count as a multiple of three so that it can be a prime (after all are multiples of 3 prime)? Two gets all the attention for being the only even prime, but the $63^\text{rd}$ Mersenne prime is the only prime divisible by the $63^\text{rd}$ Mersenne prime. Saying it like this makes the argument sound illogical (it also makes the news of 2 being the only even prime sound illogical). If your accusation is true then there are no primes, as we can apply this argument to multiples of 3, 5, 7, 11 and even the millionth prime...

See Also

• Is 0 a prime number?
• Is 1 prime?
• Prime Numbers
• List of Common Misconceptions