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The fundamental theorem of arithmetic asserts that we can produce every composite number from a unique set of prime multiplicands, so long as none of those primes equals one.

Consequently, some mathematicians call the primes the atoms of numbers. However, we can define every number, prime and composite, by adding a unique number of ones; so the analogy between atom and the number one seems more robust than the analogy between atom and prime number does.

Why do mathematicians call the prime numbers, but not the number one, the atoms of numbers?

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$1$ is the additive atom, primes are the multiplicative atoms. –  Daniel Fischer Apr 19 at 13:32
    
@DanielFischer Thank you. –  Hal Apr 19 at 18:26
    
Additionally, the additive "periodic table" $\{1\}$ is pretty boring compared to $\{2,3,5,\dots \}$. Take that chemists, our periodic table is infinite. –  Travis Bemrose Apr 19 at 21:17

4 Answers 4

up vote 4 down vote accepted

With just one kind of atom, the world wouldn't be so grand and beautiful, would it?

Mathematicians prefer to talk about primes for the same reason that they prefer to do number theory, not with real numbers ($\Bbb R$), but with only integers ($\Bbb Z$): Only in the latter case is there something interesting to discover.

The sum of ones does not provide any interesting information about a given number $n$. While it's true that $$n = \underbrace{1+1+\dots+1}_{n}$$ we still need to know the number of ones in order to make the sum become $n$. In other words, we need $n$ in order to "construct" $n$. In any event, this is just a trivial sum. In contrast, when multiplying primes, the prime numbers themselves, when multiplied, "construct" the number $n$. To get 10, the prime factorization $2\cdot5$ contains all the information.

And this is not a trivial construction. On the contrary, this structure of primes is very interesting to observe and study, just like the atomic structure of a real-world object is interesting to observe and study. There is so much to know (and still yet to be known) about primes and prime factorization, and the concept is so fundamental to more abstract algebra.

(I also very much liked the analogy of molecules from kingW3.)

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Adding atoms is just increasing number of moles,while multiplying them makes molecules.

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You are right. But you must realize that the way $1$ generates the numbers by addition is easier to study than the way the primes generate the numbers by multiplication.

The study of the first you finish by elementary school. The other not so fast.

As you said, the generation of the numbers by the one by addition is more robust. That is why it is often used as a way to define the numbers. Almost never you find a definition of the numbers beginning form the primes.

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The word "atom" is not ideal for prime numbers. Atom literally means "indivisible", so in that sense, the number 1 is as hard to divide into factors as $2,3,5\dots$ (and incidentally, the ancient Greeks regarded 1 as prime number).

However, the modern view prioritizes the algebraic structure, so the better word would be "generator" instead of atoms, and here, you can tell that 1 does not contribute anything to a product.

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