The importance of the prime numbers arise by the wide applications of the prime numbers , for details see the following

https://math.stackexchange.com/

They can be used, for example, in coding theory.

Some practical applications is known as public key cryptography, widely used to protect electronic data. These public-key encryption systems take advantage of the fact that while it is easy to multiply two huge prime numbers, there is no way to do the reverse quickly: separate a huge number into its prime factors. The prime numbers are essential in computer security programs, very important today to perform certain operations over the internet, payments with credit cards, etc. The larger the prime number used, the harder it is to decipher the code. So finding "giant" prime numbers is a current problem in which thousands of people are involved.

Look at `Peter Plichta, Das Primzahlkreuz´ and you will get more answers.

`1´ is missing in your sequence, if you use the rule you cited.

The ultimate relevance of prime numbers reside on fundamental problems even more than in applications. Prime numbers are the "atoms", or basic units of natural numbers. This follows from the fact that any natural number has a unique decomposition in prime factors. Ergo, prime numbers have a key role in number theory. Furthermore, the close relation between the distribution of prime numbers and the Riemann zeta function, provides a link to a whole world of important problems in mathematics, the most famous being Riemann's conjecture, one of the milenium problems in mathematics.

@ Reinaldo Garcia-Garcia

That´s where the problems come from.

If there is an expression in physics (note, physics is the science to define measure and unit, not math) named quantity, it is to be depicted by three symbols:

a [b] c [a = number (of amount), b = measure, c = unit]

If mathematics should have to do something with reality it must be isomorphic to this major of science for defining quantities.

So any expression in math may have numbers for a and [b], but also all the three symbols (in the same meaning). An incommensurable quantity has it´s own measure (π for example).

a is element of IN for the naturals, [b] = 1 as the measure (in math) and c = number (of amount).

Equal to units in physics are sets in math.

Because prime numbers are not equal one to another (2[1] c = 2, 3[1] c = 3, ...) they couldn´t be elements of the same set if they are not equal in quality. Two prime numbers in the set of primes are in addition:

2[piece] primes ; not 2 + 3 = 5 e.g. .

They are no atoms because they are the several times of amount of the measure of the naturals. They never could be units.

And so on.

@ Peter Kepps,

Saying that primes are the "atoms" or basic units of natural numbers is, of course a rethoric image that you have completely taken out of context, because what I meant by that expresion was clearly stated by my answer. Basic units means "fundamental, or irreducible, or indivisible elements". I think it is clear what I mean, even more since I'm invoking factorization into primes.

@ Peter Kepps, by the way, some previous answer you gave is incorrect. 1 is not a prime number. It has been a while since it was taken out of the list for consistency with other properties of prime numbers. For instance, if you allow 1 to be a prime, factorization into prime numbers is not unique. This is not the only reason, but is a heavy one, though.

@ Reinaldo Garcia-Garcia

O.k. , but by `atom´ — in science — one should not use it as to mean what one want it to mean.

I told you how to construct a prime by `atoms´ (the fundamental element of the naturals). So a prime never could be an atom. Stay clear.

To the topic of `1 belongs to the primes´ there is an actual discussion on RG. Look at: Jack Son, The number 1 is neither prime nor composite, why?

1 fulfills the definition `is only devidable by one or itself´, so it´s in.

Other definitions have to exclude 1 in their circumstances and we can go on without contradictions.

Regards Peter Kepp

@ Peter Kepps

I still don't understand your concern with the term "atom". Who has said that atoms (not quanta) are all the same? Is in that sense that the expression is used, by the analogy with *chemistry*. Complex substances (composite numbers) are made of basic units or atoms (prime numbers) which of course do not need to be all equal, the point is that they are "irreducible" or "minimal". So, I don't know what do you mean by "science"? Is chemistry a science?

Atoms and quanta are not the same.

Regards,

R.G.G.

@ Reinaldo Garcia-Garcia

In your way to use `atom´ for prime numbers it would be better to say `molecule´. Prime numbers are built by a sum of natural numbers (0 + 1, 1 + 1, 2 + 1 = 1 + 1 + 1, ...). In relationship to the origin meaning of `atom´ we see that an atom of gold is not the combination of two other atoms.

@ Peter Kepps

I can't believe we are still discussing. Why do you use sums when I'm talking about factorization? Don't you see the analogy between prime factorization and chemical formulas? Why to complicate things innecessariy by adding to what I've said more than what I've said? Don't pretend you know the way I use "atom" to later represent it in a way which is clearly not the correct one. I'm talking here about PRODUCT decomposition in prime factors which are the irreducible units because cannot be further factorized.

@ Reinaldo Garcia-Garcia

I don´t talk about the factorization of natural numbers by primes.

I only said that any prime could not been seen as not beeing built of a `more-times-addition´ of the smallest natural number > 0 the number `1´.

By that the prime `2´ contains less naturals `1´ than the prime `3´.

Give a full isometry between classes of numbers (naturals, primes) and terms like `atom´, `molecule´, `electron´, `proton´, ... then we can talk about it.

Near to my theory it is to get somethig new (only) by multiplication as you used it in factorization by primes.

If the naturals only could be built (in factorization) by the primes (but that´s not true), so by what a rule are the primes different in amount?

@ Reinaldo Garcia-Garcia

RGG:

”Prime numbers are the "atoms", or basic units of natural numbers. This follows from the fact that any natural number has a unique decomposition in prime factors.”

One (1) is out of the primes, as you said.

If any natural number has a unique decomposition in prime factors, so what are the prime factors for `1´?

If you can´t find them there is a contradiction. Or the conclusion is, that `1´ must be a prime.

Please think before post. Your comment is welcome! One may correct himself if been mistaken (it doesn´t hurt!).

Regards, Peter Kepp

@Peter Kepps:

"I don´t talk about the factorization of natural numbers by primes.

I only said that any prime could not been seen as not beeing built of a `more-times-addition´ of the smallest natural number > 0 the number `1´."

The point is that this is NOT about what you said since you were "correcting" MY answer in the first place. So you used my message to impose your agenda, which has nothing to do with what I said.

"If the naturals only could be built (in factorization) by the primes (but that´s not true), so by what a rule are the primes different in amount?"

Please, tell me when did I say that the naturals can only be built in factorization by the primes. Just tell me. This looks like you're trolling. What I have said (again) is that prime numbers are the "atoms" of natural numbers in the sense that any composite natural number has a unique decomposition into primes, and primes cannot be divided further.

"Give a full isometry between classes of numbers (naturals, primes) and terms like `atom´, `molecule´, `electron´, `proton´, ... then we can talk about it."

If I explicitly said that the analogy is with chemistry, what are protons doing here? Composite numbers are complex elements, primes are basic elements. Am I so bad at making myself clear?

"One (1) is out of the primes, as you said.

If any natural number has a unique decomposition in prime factors, so what are the prime factors for `1´?

If you can´t find them there is a contradiction. Or the conclusion is, that `1´ must be a prime.

Please think before post. Your comment is welcome! One may correct himself if been mistaken (it doesn´t hurt!)."

It is clear that if primes start at 2, then all natural numbers have a unique decomposition into primes, EXCEPT 1, it is smaller than the smaller prime number. Why do you make such a big deal? Either you exclude 1 from the natural numbers factorizable into primes, either you include 1 into primes but then fix the teorem on decomposition somehow so that you don't loose uniqueness. At the end of the the day, you DEFINE what a prime number is, so you can include 1 or not in the definition. My point is that there is some level of consensus in defining primes without the 1.

This is NOT a fundamental issue. Your concern with "wording" (more than with ideas) is unjustified.

As a final comment, this space has been completely monopolized and the original question has gone ignored. Let's please be polite and give this space back to the person who posted the original question. I imagine that you are busy doing actual science, as I am, we cannot spend all this time discussing about something as shallow as a terminology or "image" that is even used by many mathematicians as well. And as an advice, when discussing scientific ideas one needs to be polite. It is shocking that you need to ask people to "be precise", to "think before post", to provide conditions and only "then talk", and what is worst, you even assume you need to explain to a physicists what an atom is, and what science is.

Regards,

RGG

@ Reinaldo Garcia-Garcia

RGG:

” … any natural number has a unique decomposition in prime factors …”

” … in the sense that any c o m p o s i t e natural number has a unique decomposition into primes …”

”Please, tell me when did I say that …”

Please read your own answers. You are correcting yourself.

If anyone has read Plichta he may talk about the importance — that´s the question. Leave myones — we would not come to an end.

Best regards, Peter Kepp