Wikipedia:Reference desk/Archives/Mathematics/2022 September 17

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September 17

Why do some people consider 1 a prime number??

We know 1 is neither prime nor composite. But the POV that 1 is a prime number still exists and why do some people still use it?? Georgia guy (talk) 13:58, 17 September 2022 (UTC)

@Georgia guy: You should ask those people, possibly. If I had to guess I'd say they probably learned a wrong description of the notion as 'a number which divides only by 1 and by itself'. The number 1 satisfies this requirement hence the believers consider 1 to be prime. --CiaPan (talk) 15:36, 17 September 2022 (UTC)

(edit conflict) It is purely a matter of the definition, which is as it is because it is mathematically the most convenient one to use in the fundamental theorem of arithmetic. A slightly careless but commonly used definition is that a number is a prime number if it is divisible only by 1 and itself.^[1][2][3] The number 1 is not divisible by other numbers than 1 and itself, so it fits this definition. If this is how people have learned, or remember, the definition, it makes sense that they consider 1 a prime. The reason it is explicitly excluded in the definition of prime number is that if it is allowed in, the uniqueness of factorizations no longer holds: 4 = 2 × 2 = 1 × 2 × 2 = 1 × 1 × 2 × 2. One could have made a different choice, including 1 in the pantheon of prime numbers, while formulating the fundamental theorem of arithmetic as "every positive natural number is the product of a unique multiset of prime numbers that are larger than 1". By the way, our article Fundamental theorem of arithmetic confines its applicability to integers greater than 1, but 1 too is the product of a unique multiset of primes: the empty multiset.  --Lambiam 16:03, 17 September 2022 (UTC)

100 years or so ago, I think they did consider 1 as prime. But then it is an exception in many theorems. It is bad enough that 2 is prime. :-)

Already in Euclid's Elements (Book 7, Definitions), the number 1 was not a prime. Called a "unit" (μονάς – monás), it was not even considered a number (ἀριθμός – arithmós), let alone a prime number (πρῶτος ἀριθμός – prôtos arithmós). To qualify as a number, an entity had to consist of a multitude (πλῆθος – plêthos) of units.  --Lambiam 07:22, 21 September 2022 (UTC)