At the beginning, all definitions are kind of idiosyncratic, if I may say so. Only time can show whether one definition is "better" than another. Perhaps not everybody knows that the current consensus to exclude 1 is not so old : even if Gauss, in his Disquisitiones Mathematicae (1801), was perhaps the first prominent mathematician to advocate this banishment, disagrement persisted at least until the 1930's, as shown in Hardy's book A Course of Pure Mathematics (1933), where the proof by contradiction of Euclid's theorem (on the existence of infinitely many primes) begins by writing down a sequence of primes 1, 2, 3,..., N. For details, see the very interesting historical note by Caldwell & Xiong in the appended file.

This being said, what justifies the side-lining of 1 ? Number theorists largely agree that Gauss' Disquisitiones "laid the ground

work for making unique factorization central to our understanding of integers". But uniqueness of factorization obviously fails if 1 is allowed to be a prime. Note that even Euclid's proof fails if one does not take care to exclude 1 in the crucial step of the argument : if there were only a finite number of primes, then their product + 1 would have no prime factor, etc.

Goldbach's conjecture provides another example from the additive theory of primes. Actually there are 2 conjectures : "Any odd integers is the sum of at most 2 primes; any even integer is the sum of at most 3 primes". If 1 is allowed, the 2 conjectures become equivalent, whereas they actually seem to present different types of difficulties. Of course things get back to normal when adding the magic sentence "for primes not equal to 1". But reasonably, why open the door to a guy just to throw him out of the window at the first opportunity ?

Another legacy of Disquisitiones is the study of division (and possibly existence and uniqueness of factorization, UFD for short) in the rings of integers of number fields, not just Z and Q. Then the same problem as for 1 pops up when dealing with the units (= invertible elements) of the ring. Gauss himself introduced the so called ring of Gaussian integers Z [i], which he showed to possess an euclidian division, from which he derived such applications as a proof of Fermat's "two squares theorem". Unfortunately not all rings of integers are UFD. Or fortunately so, because to remedy to this default, Kummer and Dedekind invented the notion of ideals and prime ideals, which allowed to recover the UF property, but this time for ideals. This was the birth of algebraic number theory. Note that the same problem pops up again, 1 being replaced by the ring itself.

Dear Jack Son,

if 1 is defined as a prime/composite number, what would you prefer? Why? How the definition should be formulated so that it is correct?

You can find the answer at

https://www.quora.com/Why-is-1-neither-prime-nor-composite

and

https://primes.utm.edu/notes/faq/one.html

Does not meet the definition of the prime number and composite

Dr Breure has explained it beautifully !!!

http://mathworld.wolfram.com/CompositeNumber.html

Assuming 1 to be a prime leads to include it when you (try to) present any positive integer as a product of primes. It is popular demand that this decomposition is unique in this sense that number of primes and their multiplicities are unambiguously determined. E.g. 24 = 2*2*2*3=2*3*2*2 etc., since multiplication is commutative. But, there are always factors 2,2,2,3 in any order (it is a NONORDERED SET of factors, not, e.g., a sequence). If 1 can be a prime factor, this factorization is not unique: 24=1*1*2*2*2*3=2*2*2*3*1*1*...*1 = ... The 1 (or, in general, the unit of a ring) plays very,, very important role, so it is enough for a single number (element). Similary, presenting a positive integer as a sum we always use positive integers, due to unambiguity (among others), e.g. 7=3+4+0+...+0 is not what we like (in a general case)...

Dear Jack Son,

the right topic — the wrong statement.

Do not ask, say it is (in your opinion)!

[Look at Peter Plichta, Das Primzahlkreuz; and see the difference in use of primes]

First (in history) mathematicians defined that a prime `a´ is a number which only could be devided by one ore itself (a/1 or a/a).

Let `a´ be 3: 3/1 or 3/3 is true [a/1 = a; a/a = 1].

So `a´ could be used two times. Also in a = 1: a/1 or a/a => 1/1 or 1/1!

It is a definition to exclude `1´.

The fault, to name 1 not as prime, should be covered in such miracles like Peter Breuer and Wojciech Florek told you.

But `prime´ only is a definition.

They use terms like `element´, `unit´, `division´ and `ring´ without knowing the contradiction by these expressions.

Have a look at www.mathe-neu.de `Logik des Formalismus´. In the English part you will find a translation of my first paper there.

Here you may have a view on the parallel-problem to multiplication in addition. Then you will understand why such arguments, like Breuer did, are without any consequence.

At the beginning, all definitions are kind of idiosyncratic, if I may say so. Only time can show whether one definition is "better" than another. Perhaps not everybody knows that the current consensus to exclude 1 is not so old : even if Gauss, in his Disquisitiones Mathematicae (1801), was perhaps the first prominent mathematician to advocate this banishment, disagrement persisted at least until the 1930's, as shown in Hardy's book A Course of Pure Mathematics (1933), where the proof by contradiction of Euclid's theorem (on the existence of infinitely many primes) begins by writing down a sequence of primes 1, 2, 3,..., N. For details, see the very interesting historical note by Caldwell & Xiong in the appended file.

This being said, what justifies the side-lining of 1 ? Number theorists largely agree that Gauss' Disquisitiones "laid the ground

work for making unique factorization central to our understanding of integers". But uniqueness of factorization obviously fails if 1 is allowed to be a prime. Note that even Euclid's proof fails if one does not take care to exclude 1 in the crucial step of the argument : if there were only a finite number of primes, then their product + 1 would have no prime factor, etc.

Goldbach's conjecture provides another example from the additive theory of primes. Actually there are 2 conjectures : "Any odd integers is the sum of at most 2 primes; any even integer is the sum of at most 3 primes". If 1 is allowed, the 2 conjectures become equivalent, whereas they actually seem to present different types of difficulties. Of course things get back to normal when adding the magic sentence "for primes not equal to 1". But reasonably, why open the door to a guy just to throw him out of the window at the first opportunity ?

Another legacy of Disquisitiones is the study of division (and possibly existence and uniqueness of factorization, UFD for short) in the rings of integers of number fields, not just Z and Q. Then the same problem as for 1 pops up when dealing with the units (= invertible elements) of the ring. Gauss himself introduced the so called ring of Gaussian integers Z [i], which he showed to possess an euclidian division, from which he derived such applications as a proof of Fermat's "two squares theorem". Unfortunately not all rings of integers are UFD. Or fortunately so, because to remedy to this default, Kummer and Dedekind invented the notion of ideals and prime ideals, which allowed to recover the UF property, but this time for ideals. This was the birth of algebraic number theory. Note that the same problem pops up again, 1 being replaced by the ring itself.

Good question

A prime number is defined as a number with two positive divisors only. As 1 has only one positive divisor (1 itself), it does not meet the criteria to be a prime.

Similarly, a composite number is a number that can be rewritten as the product of two integers both larger than 1, and 1 does not meet that criteria too.

Hence 1 is neither prime nor composite.

Does not meet the definition of the prime number and composite

See

https://www.quora.com/Why-is-1-neither-prime-nor-composite

http://mathworld.wolfram.com/CompositeNumber.html

Thanks

The number 1 is neither prime nor composite. Why?

Answer: By Convention.

Of course, this answer is in the Euclidean domain z +.

1 = (-i)(i) can be factored into a product of 2 units over the Gaussian integers, where the units are +1,-1, +i , -i.

Best

We may define a property (German: Eigenschaft).

Thera are some useful signs in mathematics like … , -1, 0, 1, 2, 3, … .

[sorry, correction: `useful signs´ => `useful symbols´; 2018, 26 July]

Those are the numbers — of amount — (German: Anzahlen) of the measure (German: Maß) of a thing / a unit (Einheit).

[Philosophy may help why `0´ is a number of amount without calling anything]

All three of them (number, measure, unit) together form a quantity (German: Größe).

[Why English is not unambiguous in that?]

Quantities of the same unit may be combined (by other types of signs (corr.: symbols) which stay for addition, multiplication, … ).

There is a definition which is valid for all numbers of amount: the rule for primes. The number `1´ fulfills the rule!

Other definitions — who ever did them — couldn´t role that out. If I am using the definition that all integers are in the line … , -1, 0, 1, 2, 3, … and give a second definition like:

If a) real + real = r +

and b) real * real = r *

and c) real ^ real = r power [for identic reals used in `+´, `*´, `^´]

then r + ≠ r power and r * ≠ r power.

I could say — like excluding 1 from the primes — that `2´ isn´t a number.

So what is wrong? The first definition or the second one?

Where do I have to exclude `2´?

To Issam Kaddoura:

There are no units +i and –i.

The square root of minus one is solved by me. It´s the reform of mathematics. Have a look on my homepage.

Thank You Dear Professors.

@peter Kepp

You said

[There are no units -i and +i ] and

[The square root of minus one is solved by me. It´s the reform of mathematics.]

What? Are you sure?

Do you know what Integral domains are? Euclidean domains?Unique factorization domains? Modular arithmetic? Elliptic curves? Gaussian Integers? Units? unitary divisors? Generalizations of Euler function?

P-adic integers? Cyclomatic integers?? Unique factorization domains?

Do you hear the proof of last Fermat's theorem? Etc.

Let me tell you something about the Euclidean domain of Gaussian integers:

Z[i]={ a + ib, a,b in Z }

The nunits are 1 , -1 , i , -i .

Primes in Z are not necessary primes in Z[i]

5 is prime in Z but 5 = (1-2i)(1+2i) is not prime in Z[i]

7 is a prime in Z and it is prime in Z[i].

I invite you to new horizons in analytic number theory.

Any introductory textbook in this field can help.

Best wishes

@Peter Keep,

if r=1 then "r *=r power". Or not?

@ Viera Cernanova

[if r = 1 then "r * = r power". Or not?]

Yes, but not r + too. And that is, what the `and´ stands for.

r + ≠ r * ≠ r power

[all `≠´ (unequals) have to be valid]

(r = 1 => r + = 2, r * = 1, r power = 1)

(r = 2 => r + = 4, r * = 4, r power = 4)

Now o.k.?

@ Issam Kaddoura

Answer will take a little longer.

@ Issam Kaddoura

I try to do it as short as I could and as best in English as I could.

I said reform!

All terms in mathematics have to be composed by this three expressions:

1. Number (of amount) [reals]

2. Measure (in convention) [to put in square-brackets; [m] e.g.]

3. Unit (is to use definitely) [length, width, hight e.g.]

Together they express a quantity. If you don´t you may play in a science which nothing has to do with reality.

The square-root is defined as the inverse operation of the multiplication of two identical factors (in old tradition).

The identity has to be limited by amount — not to be used for the signs (prefixes)! This is correct because the prefix is convention.

If you look for two factors which develop 36 e.g., you will find:

36 I 1, 18 I 2, 12 I 3, 9 I 4, 6 I 6 (in old tradition as negatives too).

With 6 I 6 you got a minimum; the minimum of the absolute of the sum of both factors. That makes unambiguous and twice in result (but remember: t w o factors). We may look (after reform) for the minimum of the sum of the absolutes of the factors.

So yet the square-root looks for the minimum of two factors which may have different prefixes.

In reality each product (a new unit) is a result of two different units.

[We have to distinguish between the so called multiplication in the set — it´s only more-time-addition — and the origin multiplication, between the sets, producing a new one.]

The square-root of minus one never before (before my solution) has been solved. To put a cryptic sign as a result is not to bring an equivalent by solution. It´s a definition and should be written as: sqrt(–1) := i .

But if you know my reform you know how to solve the problem `sqrt(–1)´.

Using only one prefix (classical), you will get +1 and –1 or –1 and +1. By using full the reform there is no doubt about it. You only can take the square-root if there is a result of two factors and such a product must have two prefixes.

sqrt(+ –1) = +1 and –1 sqrt(– +1) = –1 and +1

You are invited to discuss the reform.

Regards, Peter

@ Peter Breuer

If you meant me in your last post

(`There are no numbers ... What say you?´)

so let me say: "That´s not my level to talk to another."

If you are able to dismount from your sarcastic horse, you are welcome.

Regards, Peter Kepp

Dear professors, thanks for your contributions. Please share your thoughts with concern.

@Peter Breuer

I agree with you about Kronecker's aphorism on the "divine gift" of the natural numbers. But as a number theorist, I partly disagree with your view of the "artificialities" (unless I misinterpreted this word) later "constructed" by man. Retrospectively, the passage from N (the natural integers) to Z (the signed integers) , i.e. from addition to substraction, may look "natural", but it was certainly not, considering the time it took to discover 0, and even the ancient Greek mathematicians did not have at their disposal the necessary no(ta)tions to really do algebra. If we agree that the fundamental algebraic operations are + and - , and their inverse operations, x and / (this includes raising to the powers and the inverse operation, taking radicals), then we can say that the Greeks knew Z and Q (the rational numbers), but in the absence of adequate no(ta)tions, they favoured the geometric point of view : Z can be represented by a scaled ruler, and Q can be constructed from two intersecting lines using Thales' intercept theorem. This explains in particular their obsession with problems of "construction with ruler and compass", which nowadays we consider as absurdly gratuitous. As for the irrational numbers (in particular radicals) just recall the discovery of the square root of 2 by the Pythagoras school, or Zeno's paradox of the arrow (see appended file). At this step of our (hi)story of numbers, we can roughly say that the natural setting for the study of numbers seems to be the theory of fields, in particular Galois theory. But this is still mainly algebra, not arithmetic, which we can define - in the same sense as in the Gauss Disquisitiones - as the study of primes.

Zeno's paradoxes lead us to the construction of R (the real numbers), and hence the formalization of the notion of convergence (see appended file). From this strict point of view, I favor the construction of real numbers by (equivalence classes of) Cauchy sequences, over what you call "partitions", i.e. Dedekind "cuts", which introduce the extra notion of order (not needed here). At this stage, it could seem as if we are leaving algebra for analysis. Not quite so. There is indeed an analytic number theory (over C) which applies in particular to the asymptotic distribution of primes via the study of special meromorphic functions such as the Riemann zeta function (over Q) or the Dedekind zeta function (over any number field). But the algebraic number theory - from Kummer on, in pursuit of Fermat's LT - also rests on an analytic foundation, not in the strict sense of convergence, but in the sense of a "completion" à la Cauchy which reveals the profound unity of mathematics (as the French say : "C'est ici que les Athéniens s'atteignent"). The completion of Q w.r.t. the usual (or archimedean) absolute value is the field of real numbers R. For any prime number p (not 1 !), one can define a p - adic absolute value (for details, see appended file), w.r.t. which the completion of Q is the field Qp of p - adic numbers. These constructions can be carried over any number field, but let us stick to Q for simplicity. You say, Peter, the p - adic numbers are "as alike as peas in a pod". What do you mean ? For two different primes p and q, Qp and Qq cannot be homeomorphic. When comparing analytically R and Qp , the situation is even worse : one is connected, the other totally disconnected. But from the number theoretic perspective, the whole family consisting of R and all the Qp is certainly united. The p-adic numbers are not "ill named", as Ostrowski’s theorem shows : up to certain equivalence relations, the archimedean and the p - adic absolute values are the only ones on Q which are compatible with the field structure of Q. Even better, since at least the 19th century, the so called global-local principle has taken root in number theory, which roughly consists, given a certain (global) arithmetic property in a number field K, in studying the analogous (local) properties in all the archimedean and p - adic completions of K (where things are easier, but disparate), and then coming back to K. My favorite comparison is with Plato's metaphor of the cave : we are in a cave, turning our backs to the sun, and what we see are just shadows projected on the walls; to achieve total comprehension, we must turn around and face the archetype behind us. Examples are cited in the appended file, but the two most extraordinary achievements up to now have been : the construction of the so called abelian class field theory , from Gauss to the first part of the 20-th century, and the pursuit of a non abelian CFT , of which Wiles' proof of the Shimura-Taniyama-Weil conjecture (which implies FLT) is a first milestone ./.

@ Jack Son

To stay in concern is a good idea. It was prime / composite not whole the number-theory you asked for.

So there is nothing else to say to Peter Breuer which stays in connection to your question.

Everybody who wants to discuss my theory (of reformation) is invited to my homepage.

But what contribution gets your like an why? You asked, so please give a vote. That could be a reason to have a deeper look on that topic (in concern).

Best regards Peter Kepp

Definition of prime number is that a positive number which has only 2 factors, ie 1 and the number itself.

Definition of composite number is that a positive number which has atleast one positive factor other than 1 and the number itself.

As you can see, 1 doesn't satisfies the basic Definition of prime and composite number. Hence 1 is neither prime nor composite.

By Shyam Singh.

Regards!

@ Adriana Santos-Caballero

Since when is 1 x 1 ≠ 1 ? And the second factor (= 1) ≠ 1?

To Adriana and Peter, especially. There is a small, but important, difference between factors and divisors. [Note: Both, factors and divisors, can be prime or composite, this is why we speak, e.g., "prime factor".] 1 is a divisor of any integer, but it is not a factor. Because we demand UNIQUE factorization. Releasing this restriction would (probably) allow include 1 into primes, but we would rather prefer to have the unique factorization than to have 1 being prime. Of course, it can be considered as convention merely. So, small correction do Adriana's answer: Prime is an integer p with exactly TWO (trivial) DIVISORS 1 and p. 1 has only one divisor, but is a divisor (not a factor!) of any number. As long as we demand that factorization means that a number n>0 can be written as a product of two numbers, say n=a*b, with a,b>1. OK. Let's assume that 1 is a prime and we allow factorizations with (a lot of) 1's. E.g., 6=1*6=1*1*..*1*2*3 etc. Then we can introduce an equivalency: factorizations with different numbers of factors equal 1 are equivalent. So two factorizations 6=2*3 and 6=1*2*3 are equivalent and as a representative we choose this with no 1's (the least number of 1's, in a general case) . If we demand that factorization means at least two factors then we extend the factorization problem to some trivial ones: n=1*n, including p=1*p and 1=1*1 (this is the least number of 1's with TWO factors in total). What does it change? Practically nothing. We are interested in non-trivial factorizations, i.e. these with no 1's, and the RSA code is still difficult to break... Summarizing: OK, let 1 be a prime and factorizations n=1*n are allowed (with the equivalency defined above). Since these factorizations are known then we are again have to cope with a factorization "sub-problem": for a given n>1 find two numbers a,b>1 such that n=a*b. Note that knowing a and b we know the whole equivalency class of "different" factorizations [{a,b}]={n=a*b, n=1*a*b, n=1*1*a*b, ...}. Are you happy with this? 1 is a prime in this approach! BTW, in this approach a number is composite if it has non-trivial factorization, i.e. into a product of two numbers greater than 1, so 1 is not composite.

29th Answer

@ Wojciech Florek

There is nothing which leads to a wrong path in what you say.

But:

First (in history?) rule to prime was `… could be devided by one or itself´.

[nothing about number (of amount) of divisors]

Later came such rules like `integer … two trivial divisors´.

So exclude `1´ from all the fallowing rules and `1´ could be prime without any contradiction.

Could you be happy with this?

One is neither a prime nor a composite number. A prime number is one with exactly two positive divisors, itself and one. One has only one positive divisor. It cannot be written as a product of two factors, neither of which is itself, so one is also not composite. It falls in a class of numbers called units. These are the numbers whose reciprocals are also whole numbers

@ Boushra Youssif

Please look at physics. That one which could be taken there, for more times, is the [measure]; not the unit!

For example we write: 3 [m] length (`length´ is unit).

If mathematics should not only be art, so there the definitions have to be the same. I did a reform which takes definated expressions to their right place.

A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive factors, 1 and the number itself. Natural numbers that have more than two positive factors are called composite. 1 has only one positive factor i.e. no.1 only. Hence 1 is neither prime nor composite

One is primary to or precedes all primeS (is necessarily defined before the primes, encompassing them as so many ones anyway; not proper or circular reasoning to define (de-fine=of the finite) something with itself), if not primary to all, the Prime of primes, thus not among the primeS per se or for that matter, composites.

@ Cj Nev

By theory of language it is forbidden to define something by itself (a prime is a prime; i.e.).

Here you are right.

Cj Nev: … defined with itself …

Here you are wrong.

The definition goes like: `prime is, if a natural number only …´.

[definition of a property]

The number `one´ isn´t the definition of a property! It is the first (included) natural number.

No one said: 1 = 1 or `prime means …´ = `prime means…´ . If you want to argue by correct logic you have to hold the rule for depiction (function).

Could I help you?

Cj Nev: One is primary to or precedes all primeS … .

Assertions need a proof.

Cj Nev: ... (de-fine=of the finite) ...

What a source tells that?

The definition of prime number requires that it is an integer at least 2. Thus, 1 cannot be prime. A composite number must have at least two (not necessarily distinct) prime factors. Thus 1 cannot be composite either.

That being said, there are some good reasons to exclude 1 from the set of primes. Namely, there are several big theorems in Number Theory that wouldn't work if 1 was a prime (e.g., The Fundamental Theorem of Arithmetic).

@ Robert A. Beeler

Please look some posts above.

First (in history) there was the definition of prime: `to be divisible by one or itself´

Later came such definitions like: `theorem of arithmetic´, ...

So, exclude the number one doing that definitions and everthing is alright.

Do you understand?

The primes don´t built a set! An element of a set must be equal to another element of the same set. Different things may be of the same category but never of the same set. The opposite opinion is a cardinal-fault of expert opinion of mathematics and leads to many contradictions.

Please excuse me!

One should know what one asserts. So I did not.

The first (is it the first?) definition of `prime´, done by Euclid (following the school of Platon) in his `Elements´ could be interpreted as: `primes are those numbers, which are devidable only by `one´; excluding the `one´ itself.

I had a look at the translated text on: www.antike-griechische.de/Euklid.pdf

There is no word like `dividable´ — only a relationship basing on `to measure´ (= find the factor — element naturals — for the `unit´.

[By my reform `unit´ must be named as `measure´.]

Euclid´s Book VII (in ` [ ]´ explanations to needful corrections leading to my reform):

1.: The measure [not the unit] is the atomar thing of selves; a single part of all equals [of

the unit].

2.: A number has to contain more than one measure [not unit].

3/5.: a is a devisor of b if a measures b correct (without any rest) so n-times a is equal b.

[Note: If a number is the measure of another number it must be a more times addition of it (more than

only one).]

11. Prime is a number which only could be measured by the measure.

Problems by the definitions of Euclid in relationship to todays use:

1. Primes couldn´t be measured by themselves (contradiction in modern definition).

2. The `1´ isn´t a number (a number of amount, bad English!) in Euclid´s definitions,

because it means `single´. [Tip: Don´t do number theory using English!]

3. Because every measure (at Euclid´s definitions) leads to a more times (n-times) addition,

there are no results for 1/x (Euclid, Book V).

Knowing all that I prefer to use the `1´ as a prime and to exclude `1´ at other definitions where it doesn´t lead to unambiguous results.

To avoid trivial cases in many important results of commutative algebra and number theory, primes are assumed non-units (by definition) in every commutative ring. Therefore the fact that 1 is not considered a prime number in the integers is a particular case of the general fact that units are not considered primes (or irreducibles) in any commutative ring.

Indeed, notice that if we stipulated (by definition) that 1 is prime, then we would automatically lose the uniqueness of prime factorizations. And, on the other hand, it doesn't seem to be a significant advantage on assuming that units are primes by definition.

@ Felix Gotti

Please give statements on whole the discussion.

To pick up only one point seems to be the same fault as to follow Euklid only by that definitions which seem to confirm.

Again and again: Exclude the `one´ at other definitions!