# Isn't 2 a vacuously prime number?

Since there are no numbers between two and one 1, 2, two cannot be divisible by a number other than itself and one. This is an instance of what mathematicians call vacuous truth, where something is true simply because no potential counterexample can exist. They still consider such statements as true. but since a vacuously true truth is vacuously true. doesn't it make 2 a vacuously prime number?

• There are no "vacuously prime numbers". – Mauro ALLEGRANZA Mar 28 '18 at 7:14
• For an "alternative" def, you can see Euclid's Elements, Book VII, Def.11: "A prime number is that which is measured by a unit alone." This means that n is prime iff there is no k > 1 such that n is multiple of k. Thus, 7= 7 x 1 is prime and 2=2 x 1 is prime. – Mauro ALLEGRANZA Mar 28 '18 at 7:19
• The formal definition must be: "Prime(n) iff for every k, for every h (if n=kh , then either k=1 or k=n)". – Mauro ALLEGRANZA Mar 28 '18 at 7:51
• Possible duplicate of Why is 2 considered a prime number? – JeffUK Mar 28 '18 at 12:02
• @JeffUK In the first question I did not at all convinced that the number is a prime number; in the second question, I was convinced that the number of two is a prime number, but its type was not clear. – Achmed Mar 28 '18 at 16:31

## 2 Answers

One could consider 2 a “vacuous” prime number if one defines the word “vacuous” to refer to a natural number having the set of its possible divisors greater than 1 but less than that natural number as empty. More simply for natural numbers, it would be the set of natural numbers greater than 1 and less than that natural number. If the set is empty then call the natural number “vacuous”. Then 2 would be the only vacuous prime number in the set of natural numbers, but that is not a problem.

What is more troubling is that 2 would also be the only vacuous natural number even if one considered composite numbers. No other natural number would be “vacuous” since every other natural number has a non-empty set of possible divisors that are less than the number and greater than 1. This makes me wonder how useful the idea of “vacuous” is. Is it only an idea that makes sense for 2?

Contrast the idea of “vacuous” with the idea of “even”. The natural number 2 is also the only “even” prime number. What makes “even” promising and potentially more useful than “vacuous” is that there are other even natural numbers. The idea of “even” not only works with 2 although these other even natural numbers are composites. The idea of “even” is useful beyond the number 2.

Although one could come up with a definition for a vacuous natural number, that it might apply only to the natural number 2 and not to any other natural number restricts its usefulness.

• > This makes me wonder how useful the idea of “vacuous” is? Given the definition of prime numbers, it can be said that it depends on the definition – Achmed Mar 28 '18 at 16:36
• @Achmed It itself may not be useful, but that doesn't mean it isn't worth considering. Something useful, and perhaps surprising, may come out of considering even options that one ultimately doesn't use. – Frank Hubeny Mar 28 '18 at 16:58
• If we had a word for numbers divisible by 3, the fact 2 is the only even prime would stop sounding so surprising. – Veedrac Mar 29 '18 at 2:37
• @Veedrac Good point about 3. It is really the unit 1 that needs to be handled separately to focus on the core idea of unique factorization. – Frank Hubeny Mar 29 '18 at 13:26

To disentangle the following issues:

1. Relation between truth and counterexample: A statement is true if and only if no counterexample exists. The term "potential" is superfluous.

2. Tautology: A tautology is a statement which is true independently from the truth-value of its components. Example: "X or not X" is true independently from the truth value of the variable X.

3. Vacuous truth: A statement which is true, because it refers to the empty set. Example: All kings of France in the 20th century had red hair.

The statement "2 is a prime number" is true because it satisfies the definition, see your question Why is 2 considered a prime number? The fact, that the number "2" satisfies the definition by trivial reason, has no relevance for the truth value of the statement.

"2 is a prime" is not a vacuous truth because primality of a given number is a statement about all integers: The proof shows that none pair of integers different from 1 and 2 satisfies the product relation.

Note: For any x, the non-existence of such integer factors is exactly the content of x being prime. The non-existence of these factors is not a vacuous truth.

• There is an empty set between two and one. – Achmed Mar 28 '18 at 9:03
• @Achmed Yes. But first you have to prove that only numbers between 1 and 2 are candidates for the factorization. Why not 2= xy with x,y >3? To exclude this case you need a rule how multiplication behaves with respect to ordering: Namely the rule x>2 and y>2 implies xy>2. – Jo Wehler Mar 28 '18 at 9:08