Why is 1 not considered a prime number? Or, why is

Anne Wacker

Anne Wacker

Answered question

2021-12-17

Why is 1 not considered a prime number?
Or, why is the definition of prime numbers given for integers greater than 1?

Answer & Explanation

Neil Dismukes

Neil Dismukes

Beginner2021-12-18Added 37 answers

One of the whole "points" of defining primes is to be able to uniquely and finitely prime factorize every natural number.
If 1 was prime, then this would be more or less impossible.
Elaine Verrett

Elaine Verrett

Beginner2021-12-19Added 41 answers

Its
nick1337

nick1337

Expert2021-12-28Added 777 answers

The main point of talking about prime numbers is Euclid's theorem that every positive integer can be written uniquely as a product of primes. As Justin remarks, this would break horribly if 1 were considered prime, for example we could factor 2 as 2×1×1×1×1×1. Instead we say that 1 is not a prime, but it is the product of zero primes (see Why is x0=1 except when x=0? to understand why any prime multiplied by itself 0 times is 1) so Euclid's theorem works out nicely!

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