# Natural numbers.

Perhaps I am reading it wrong.

Wolfram states that a prime number (or prime integer, often simply called a "prime" for short) is a positive integer

More concisely, a prime number
is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored.

Wikipedia states that a prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Are these examples generalizations, or is that what people really think?
These definitions are far too restrictive in the sense of what a prime really is. they are far simpler entities than that. However they can be represented neatly in the contexts that are given.

integers, Reals, naturals, Primes, ect.. Could someone give a definition that would describe what is accepted as a standard.
and if a definition exists today that is to restrictive, how would one go about generalizing it.

I have also read that the concept of prime number has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense.
This fits better in with what i know, but again why has it not been redefined in the case of prime numbers.. Restrictive barriers seem to have been placed around this concept that do not embody what it truly is as a whole.
In regards to Prime "numbers", any number can be prime.

#### topsquark

Math Team
How else would you define a prime number? That's what they are! They can be described in some sense by properties they have but in the end it all boils down to the original definition.

-Dan

#### topsquark

Math Team
How else would you define a prime number? That's what they are! They can be described in some sense by properties they have but in the end it all boils down to the original definition.

And how can 4 be prime number? Sorry but that's ridiculous statement.

-Dan

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#### SDK

Perhaps I am reading it wrong.

Wolfram states that a prime number (or prime integer, often simply called a "prime" for short) is a positive integer

More concisely, a prime number
is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored.

Wikipedia states that a prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Are these examples generalizations, or is that what people really think?
These definitions are far too restrictive in the sense of what a prime really is. they are far simpler entities than that. However they can be represented neatly in the contexts that are given.

integers, Reals, naturals, Primes, ect.. Could someone give a definition that would describe what is accepted as a standard.
and if a definition exists today that is to restrictive, how would one go about generalizing it.
Prime numbers generalize to prime ideals. If $I$ is a proper ideal of a ring, $R$, then $I$ is prime if whenever $ab \in I$, then either $a \in I$ or $b \in I$. This is a generalization of prime numbers as you normally encounter them since it is fairly easy to show that the prime ideals of $\mathbb{N}$ are precisely the ideals generated by the prime numbers. In other words, $I$ is a prime ideal of $\mathbb{N}$ if and only if $I = (p)$ for some prime number $p$.

I don't know how to parse the rest of your question. There are no primes in the reals because they form a field. It is not hard to show that a field trivially can not have any prime ideals

topsquark

#### TR-Resort

Since you like definitions so much, consider the non-mathematical definition
PRIME - (adjective) of first importance; main; principal; foremost
A prime is PRIME because it represents the first incarnation a number that is not a product of any other factors before it. That's pretty special.

#### topsquark

Math Team
Since you like definitions so much, consider the non-mathematical definition
PRIME - (adjective) of first importance; main; principal; foremost
A prime is PRIME because it represents the first incarnation a number that is not a product of any other factors before it. That's pretty special.
I understand what you are trying to say, but this is Mathematics. Non-Mathematical definitions have no meaning here.

-Dan

#### TR-Resort

I understand what you are trying to say, but this is Mathematics. Non-Mathematical definitions have no meaning here.

-Dan
The OP in another post tried to define irrational numbers as 'illogical" numbers - I was just applying his warped reasoning.

topsquark

#### mrtwhs

I can't wait until he gets to imaginary numbers. On the other hand, I believe that he thinks the complement of the natural numbers is called the unnatural numbers.