Why '1' is not a prime number?

Answer 1

Because it is not divisible by negative zero.

Answer 2

It's because 1 is classified as a "unit." Technically 1 satisfies all the requirements for being prime, but mathematicians have put units in a category separate from primes because of the Fundamental Theorem of Arithmetic.

Recall the Fundamental Theorem of Arithmetic, which says that every natural number (bigger than or equal to 2) can be expressed UNIQUELY as the product of powers of primes (try this for numbers like, say, 254 or 1972). However, if we were to include 1 as a prime, then we lose the uniqueness that I stressed above. Here's what I mean; suppose we look at the number 72. This has the unique factorization (2^3)(3^2). However, if we include 1 as a prime, then we have (2^3)(3^2) as a factorization, but also (1^2)(2^3)(3^2), (1^3)(2^3)(3^2), and so on for any power of 1. So we've lost unique factorization. This property that I'm mentioning so much (unique factorization) turns out to play a very important role in Number Theory, so mathematicians sacrificed the number 1 as a prime to maintain this property. So 1 is classified simply as a unit, and not as a prime.

Answer 3

prime nos. are not divisible by any no. expect 1 and that no.
So it can be called .
A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. (More concisely, a prime number is a positive integer having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.

Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers whose divisors are trivial and given by exactly 1 and .

The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any . In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "2 pays its way [as a prime] on balance; 1 doesn't."

Answer 4

The number 1 is a unit as is -1: they both divide 1. The difficulty is that both of these dived into *every* number evenly.

A prime number, p, is a non-unit with the property that if p divides a product ab, then p divides a or p divides b.

An irreducible is a non-unit number, p, with the property that if p is written as a product ab, then either a or b is a unit.

For integers, primes and irreducibles are the same thing, but they can be different in other number systems. Notice that the definition of irreducible is what most people think of as the definition of prime. In either case, units are excluded so 1 is neither a prime nor an irreducible.

The reason for the exclusion is ultimately the fundamental theorem of arithmetic which states that every number can be factored into primes in essentially a unique way. In this, primes that are units times each other are considered to be 'essentially the same'. So, for example 7 and -7. So 14=2*7=(-2)*(-7) is considered to be essentially the same factorization.

Answer 5

A prime number is whatever mathematicians define to be a prime number. It is just to give them a neat way of being able to talk concisely after that about "prime numbers", and about what properties they have while other numbers don't.

Among the prime numbers, 2 is sometimes rather awkward, and there are quite a lot of important properties which apply only to "the odd primes", i.e. all the primes except 2. But it isn't a big enough deal to make mathematicians want to make up any shorter snappier a term for them than "odd primes".

There are ever so many useful properties that the number 1 doesn't share with prime numbers, and hardly any that it does. So if 1 was called a prime, then there would be very little that mathematicians could say about "prime numbers", and all the rest of the time they would have to be talking about "the primes greater than 1", or "all the primes except 1", and it would just be cumbersome. So we are back to the fact that prime numbers are what mathematicians say they are.

Sometimes a mathematician is not so good at English as he is at mathematics, and his definition of prime number which was meant to exclude 1 turns out to include 1 if you analyse it down to the last comma. This doesn't prove that 1 is prime, it just proves that the mathematician is not very good at English.

Answer 6

It would not be a very useful prime ... as it would be able to tricks that other primes cannot.

1 = 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 ...

This could be considered a trivial case.

It is the same as taking a loan but with the loan amount = 0,
then when you calculate interest 10% x 0 = 0, 20% x 0 = 0, not very useful or meaningful.

Answer 7

Dear,

Prime numbers are those numbers which are completely divisible by by only two numbers i.e.. by themselves and another by 1. Now 1 is the only number which is only completely divisible by 1 only. There is no other number which can divide 1 complelety without leaving and remainder and havinig a whole number as divisor. Thus it does not satify the definition of being prime.

Further number 1 is a exclusion beacuse it is the basis of deciding the prime numbers.

All numbers are completely divisible by themselves. Apart from them 1 is the only number which completely divides each and every number completely, without an exception.

So 1 is taken as the unit of measurement or as a control parameter to satify the condition.

hence 1 is not PRIME.

Answer 8

To maintain the validity of the 'fundamental theorem of artithmetic'. This theorem says that any positive integer is UNIQUELY factorable into the product of primes raised to integer powers. Allowing 1 to be prime would violate uniqueness since multiplying by another 1 would constitute another factorization. For exampe:

51 = 3*17
51 = 3*17*1
51 = 3*17*1*1
etc
Every integer would have an infinite number of factorizations which is not cool.
Just for kicks, note that 2 is the only even prime.

Answer 9

prime number must be composed of 2 different numbers. One is "1" ;the other is itself,so "1" which is 1*1 is actually certainly by all means really of course not "prime number"
believe me.
Your sincerely,

Answer 10

because a prime number has onlt itself and one at factors, and in 1's case, this would mean it only has 1 and 1. These are both the same numbers, so it isn't prime. that's what i think anyway. hope it helps!

Answer 11

The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below we give four answers, each more technical than its precursor.

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.
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.