Why isn't 1 a prime number? Explain.

[Krakin]

Definition of a prime number, an integer that has no integral factors but itself and 1.

So if 1 divided by itself or 1 is one therefore not 1?

Of course I understand that 1 isn't a prime number but why is it not?

[Bradprobert]

If this question interests you, you might look at the history of the primaility of one as described in the papers "What is the smallest prime?" [CX2012] and "The History of the Primality of One: A Selection of Sources" [CRXK2012].

Answer One: By definition of prime!

The definition is as follows.

An integer

greater than one

is called a

prime number

if its only positive divisors (factors) are one and itself.

Clearly one is left out, but this does not really address the question "why?"

Answer Two: Because of the purpose of primes.

The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved:

The Fundamental Theorem of Arithmetic Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size.

Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1^.a = 1^.1^.a = ... That is, divisibility by one fails to provide us any information about a.

Answer Three: Because one is a unit.

Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units.

So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with the number one (and with primes).

Answer Four: By the Generalized Definition of Prime.

(See also the technical note in The prime Glossary' definition).

There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they must be found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:"

An element

p

of the ring D, nonzero

and not a unit

, is called

prime

if it can not be decomposed into factors

p

=

ab

, neither of which is a unit in D.

Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.

See, for example, the section on factoring primes in A Brief Introduction to Adelic Algebraic Number Theory.

[Jessica]

[TeamHT]

Itself "and" 1 denotes two separate numbers...which makes sense since the actual definition is an integer greater than 1.

[Jessica]

Itself "and" 1 denotes two separate numbers...which makes sense since the actual definition is an integer greater than 1.

close, watch that video for a great explanation.

[ThreehundredC]

math class funfacts

[Kyblack76]

All of you...

go get a hand job...

[boom50cal]

All of you...

go get a hand job...

lol Kyle you joker you! Don't you know nerds get all the bitches?

[Jessica]

All of you...

go get a hand job...

I give myself one all the time bitch.

[Noel989]

All of you...

go get a hand job...

I give myself one all the time bitch.

Lol