Meaning

An informal sense

The addition of smaller counting numbers: Any counting number, other than 1, can be built by adding smaller counting numbers.The only way to make a larger counting number is to multiply two smaller counting numbers.

We can multiply 9 and 4 to get 36; or we can multiply 6 and 6 to get 36; or we can multiply 18 and 2 to get 36; or we can multiply 2 * 2 * 3 * 3 to get 36.These are numbers like 10 and 36 and 49, which can be combined from smaller counting numbers to produce a composite number.

This can't be done for some numbers.In my opinion, the only way to multiply and count 7 is by multiplying it by 1.So, 7 is needed!.So we aren't really building it from smaller elements; we need it from the beginning.

It is informal to say that primes are numbers that cannot be made by multiplying other numbers.This captures the concept well, but does not provide a good enough definition, as it has too many gaps.Seven can be composed by adding two other numbers: for example, two times three.

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A formal definition

A prime number is every integer that is divisible by two distinct whole numbers, namely 1 and the number itself.

Clarifying two common confusions

There are two commonly confused terms:

The number 1 is not prime.

According to the definition, it's not allowed.As it says, "two distinct whole-number factors", the only way to express 1 as a product of whole numbers is 1 * 1, in which the factors are the same, so they are not distinct.The informal idea also rules it out because it cannot be built by multiplying whole numbers.

But why exclude it?.

It is not arbitrary to use mathematics.

3 × 44 × 31 × 121 × 1 x 122 × 61 × 1 × 1 × 2 × 6

Using 4, 6, and 12 clearly violates the restriction to be “using only prime numbers.” But what about these?

3 × 2 × 22 × 3 × 21 × 2 × 3 × 22 × 2 × 3 × 1 × 1 × 1 × 1

Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Including 1 trivializes the question. Excluding it leaves only these cases:

3 × 2 × 22 × 3 × 22 × 2 × 3

This is a much more useful result than having every number be expressible as a product of primes in an infinite number of ways, so we define prime in such a way that it excludes 1.

The number 2 is prime. Why?

Students sometimes believe that all prime numbers are odd. If one works from “patterns” alone, this is an easy slip to make, as 2 is the only exception, the only even prime. One proof: Because 2 is a divisor of every even number, every even number larger than 2 has at least three distinct positive divisors.

Another common question: “All even numbers are divisible by 2 and so they’re not prime; 2 is even, so how can it be prime?” Every whole number is divisible by itself and by 1; they are all divisible by something. But if a number is divisible only by itself and by 1, then it is prime. So, because all the other even numbers are divisible by themselves, by 1, and by 2, they are all composite (just as all the positive multiples of 3, except 3, itself, are composite).

Mathematical background

Unique prime factorization and factor trees

The question “How many different ways can a number be written as a product using only primes?” (see why 1 is not prime) becomes even more interesting if we ask ourselves whether 3 × 2 × 2 and 2 × 2 × 3 are different enough to consider them “different ways.” If we consider only the set of numbers used — in other words, if we ignore how those numbers are arranged — we come up with a remarkable, and very useful fact (provable).

Every whole number greater than 1 can be factored into a unique set of primes. There is only one set of prime factors for any whole number.

Primes and rectangles

It is possible to arrange 12 square tiles into three distinct rectangles.

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Seven square tiles can be arranged in many ways, but only one arrangement makes a rectangle.

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How many primes are there?

From 1 through 10, there are 4 primes: 2, 3, 5, and 7.From 11 through 20, there are again 4 primes: 11, 13, 17, and 19.From 21 through 30, there are only 2 primes: 23 and 29.From 31 through 40, there are again only 2 primes: 31 and 37.From 91 through 100, there is only one prime: 97.

It looks like they’re thinning out. That even seems to make sense; as numbers get bigger, there are more little building blocks from which they might be made.

Are primes ever stopped?.Imagine for a moment that they eventually do.Assume that there were a "greatest prime number" - let's call it p.As an example, if we multiply all of the prime numbers we already know (all of them from 2 to p) and then add 1 to the product, we will get a new number, q, that is not divisible by any of the prime numbers we know about.(Dividing by any of these primes would leave a remainder of 1) Thus, either q is prime itself (and certainly greater than p) or it is divisible by some prime we haven't yet encountered (that, in turn, must also be greater than p).

Let's imagine 11 is the largest prime number.

Think of 13 as the largest prime.