# can you explain why 1 is not a prime number?

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It is reasonable question to ask. After all, one of the attributes of a prime number is non-divisibility by a number other than 1 and itself, and 1 shares this property with the primes. The short answer is convenience. Now you may well ask whether it is open to mathematicians to take convenience into account when formulating definitions. Again, the short answer is ‘yes’. Formally, a definition is nothing more than an agreement on the use of words. As such we are free to use words in any way we like, provided we can persuade the mathematical community to adopt our proposals for any particular usage. But mathematicians have reasons why they define their words in the way they do, why for instance 1 is specifically excepted from their definition of prime number. The reason here is that if they didn’t do this, they would have had to endure the inconvenience and tedium of a large number of their theorems being stated as such and such is true provided that p is a prime other than 1. For instance, Fermat’s beautiful theorem (not to be confused with his little theorem and last theorem!) that every prime of the form 4r + 1 can be expressed as the sum of two integer squares in exactly one way would have had to be hedged in by the proviso ‘provided p is not 1’. There is no deeper reason than this why 1 is not counted as a prime. Note that this is not quite the same as why division by 0 is excluded from legitimate operations. The prohibition here is integrally embedded in the axiomatic structure of the real and complex numbers.

• Dixon
Lv 7
1 month agoReport

@BENARD But the two factor rule is just a post hoc definition to exclude 1 (and 0). The central idea of primes is they are not compound numbers and after that it is a matter of convenience if we wanted to include 1 (and 0) or not. So we fabricated a definition to give us the numbers we wanted.

• Proof: The definition of a prime number is a positive integer that has exactly two positive divisors. However, 1 only has one positive divisor (1 itself), so it is not prime.

• IF it were Prime, it would be the only Prime number!

• A prime number is a number that is divisible ONLY by itself AND one. Because one cannot be divided by itself AND one, it isn't prime.

Source(s): Basic Math skills
• The Greek geometers (mathematicians from 2000 years ago) had a problem with that too. They even thought of excluding 1 from the list of counting numbers just to avoid the problem (after all, you never need to "count" if you only have one of something). For the "counting" reason, zero has never been included in discussions of prime numbers.

In modern mathematical jargon, in any field (or ring, or multiplicative group, or...) where it is possible to have primes, any number that is a "unit" cannot be prime. Conversely, a prime number is a number that cannot be divided by anything else than itself or a "unit".

A unit is a number for which a reciprocal exists. A reciprocal is a number which, when multiplied by the original number, gives a product of 1.

In the integers, "1" is a unit because you can multiply it by its reciprocal (also 1) to get 1:

1 * 1 = 1

"-1" is also a unit, because its reciprocal (-1) exists in the integers.

-1 * -1 = 1

2 is not a unit because the number 1/2 (its reciprocal) does not exist in the integers.

However, in the field of rational numbers, the number 1/2 does exist and, therefore, 2 is a "unit". In the field of rationals, there cannot be any prime because each number (except 0) has a reciprocal in the same field.

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All this is, of course, more complicated than the shortcut definitions given to beginners, because you begin by dealing only with positive integers. A prime is a number, greater than 1, which cannot be divided by any other number than itself and 1. A number that CAN be divided by other numbers is called a composite number.

15 = 5*3

15 is a composite number (non-prime)

The definition involving the "unit" becomes necessary when you move into other mathematical fields, for example, Complex numbers and polynomials (yes, there can be "prime" polynomials). The idea is to identify what is a "unit", then apply the definition.

in real polynomials, 4x^2 + 4x + 4 is "prime" because it cannot be factored without using a "unit"

while 4x^2 + 8x + 4 is not prime because it can be factored as (2x + 2)(2x + 2) = (2x+2)^2

• By definition a prime number is a number that can be divided only by itself and one.

The first few prime numbers are 2,3,5,7,11,13,17______

• A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

• Because 1 is neither a prime number or composite number as it has only one factor which is itself, whereas a prime number has two factors and a composite number has more than two factors.

• A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.

• David B.
Lv 7
1 month agoReport

BERNARD; according to the definition of a prime number (or a prime) is a natural number greater than 1. This is the very first line in the explanation at the link I provided.

• No, because it is.