Some sources implement this notion by introducing the concept of a prime integer. There are advantages to this approach, because then special provision does not need to be made for negative integers. ![]() Some treatments of number theory define a prime as being either positive or negative, by specifying that a prime number is an integer with exactly $4$ integer divisors.īy this definition, a composite number is defined as an integer ( positive or negative) which is not prime and not equal to $\pm 1$. Rose: A Course on Group Theory uses $\varpi$ (a variant of $\pi$, despite its appearance) to denote a general set of primes. The letter $p$ is often used to denote a general element of $\Bbb P$, in the same way that $n$ is often used to denote a general element of $\N$.ġ978: John S. This notation is not standard (but perhaps it ought to be). Some authors use the symbol $\Bbb P$ to denote the set of all primes. The concept of primality can be applied to negative numbers as follows:Ī negative prime is an integer of the form $-p$ where $p$ is a (positive) prime number. That is, an integer greater than $1$ which is not prime is defined as composite. So, referring to an odd prime is a convenient way of specifying that a number is a prime number, but not equal to $2$.Ī composite number $c$ is a positive integer that has more than two positive divisors. Therefore, apart from $2$ itself, all primes are odd. Sloane (Ed.), 2008).Įvery even integer is divisible by $2$ (because this is the definition of even). This sequence is A000040 in the On-Line Encyclopedia of Integer Sequences (N. ( The Elements: Book $\text$: Definition $11$) Where $a$ and $b$ are both positive integers less than $p$.Ī prime number is that which is measured by an unit alone. Then $p$ is a prime number if and only if $\forall a, b \in \Z: p \divides a b \implies p \divides a$ or $p \divides b$Ī prime number $p$ is an integer greater than $1$ which cannot be written in the form: ![]() Let $p \in \N$ be an integer such that $p \ne 0$ and $p \ne \pm 1$. ![]() Where $\map \tau p$ denotes the divisor counting function of $p$.Ī prime number $p$ is an integer greater than $1$ that has no positive integer divisors other than $1$ and $p$.Ī prime number $p$ is an integer greater than $1$ that has no (positive) divisors less than itself other than $1$. Then $p$ is a prime number if and only if: Then $p$ is a prime number if and only if $p$ has exactly four integral divisors: $\pm 1$ and $\pm p$. A prime number $p$ is a positive integer that has exactly two divisors which are themselves positive integers.
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