# Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I learned that (in order to make prime factorization unique) 1 is not considered to be prime, making 2 the first prime.

## Questions:

• When and by whom was the characterization of primes as natural numbers that are only divisible by 1 and by themselves without remainder, given?

• When and by whom was 1 deprived its prime status?

• Does a definition of primes exist that rules out 1, and that does not refer to prime factorization or its uniqueness? (Would “a natural number is a prime number if it is only divisible by 1 and by itself and if it doesn’t divide bigger prime numbers” be acceptable?)

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## marked as duplicate by Benjamin Steinberg, Stefan Kohl, Goldstern, Karl Schwede, Jack HuizengaMar 6 '14 at 23:20

@JamesCranch: agreed, the question is very similar; let's focus on the last bullet asking for an alternative definition of primes –  Manfred Weis Mar 6 '14 at 13:25
See the references [CX2012], [CRXK2012] in primes.utm.edu/notes/faq/one.html. –  Dietrich Burde Mar 6 '14 at 13:26
The answer to the last question is yes, as there is a definition that works in any ring. This part of the question is not really suitable for MO, as this is standard introductory algebra (I will not pass judgement on the history part of the question as that is something I know nothing about). –  Tobias Kildetoft Mar 6 '14 at 13:41
Please learn the difference between restrictive and non-restrictive commas. –  TRiG Mar 6 '14 at 17:31

Here are two papers that survey the history of how prime numbers are defined: “What is the Smallest Prime?” by Chris K. Caldwell and Yeng Xiong (Journal of Integer Sequences, vol. 15 (2012), Article 12.9.7) and “The History of the Primality of One: A Selection of Sources” by Chris K. Caldwell, Angela Reddick, Yeng Xiong, and Wilfrid Keller (Journal of Integer Sequences, vol. 15 (2012), Article 12.9.8).

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Consider Euclid's definitions:

• "a prime number is one measured by a unit alone"
• "a number is a multitude composed of units."

(Book VII, definitions 11 and 2, for instance here.)

So, to the extent Euclid was clear on this, 1 was not a prime number because it was not a number.

I doubt we have any older texts that clearly treat 1 as a prime number. So I wouldn't say that anyone deprived 1 of its primality.

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the definitions of Euclid as listed in your answer are not very convincing to me because there are the terms "unit", "measure", "multitude" and "composed" that leave room to interpretation and might also include 1 as a prime. –  Manfred Weis Mar 6 '14 at 13:53

Even if it is not truly exciting, I like this one :

"$~p~$ is prime iff it has exactly four different divisors $~-p,-1,1,p~$."

Not considering $~1~$ as a prime number is probably simpler in view of uniqueness in prime factorization...

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The “four” in the definition is quite unmotivated. Do you want $1$ to be prime in the ring $\mathbb Z[i]$? –  Emil Jeřábek Mar 6 '14 at 18:11
Right, but make $i$ instead of $1$. –  Włodzimierz Holsztyński Mar 6 '14 at 20:24