The Prime Debate: Is 1 prime?

[Enigma_Nova]

...
I believe that it is NOT.
G_Slacker will fill you in with the chatlog

[GePap]

And now, for the award of "possibly most boring thread topic", the nominee's are:

The Prime Debate: Is 1 prime?
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[geeslaka]

Posted directly in because it's only 161 lines.

19:28 < G_Slacker> Programmers Proof that all odd numbers are prime: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 will be fixed by the next patch, 11 is prime...
19:28 < WIA_working_AFK> Catch you later...
19:29 < Enigma_Nova> I'll be in here
19:29 -!- WIA_working_AFK [~mrwhereit@131.203.9.136] has left #apolyton []
19:29 < Enigma_Nova> 1 is not prime you ditz
19:29 < G_Slacker> Yes it is.
19:29 < G_Slacker> Define prime.
19:29 < Enigma_Nova> Fundamental theorem of Arithmetic: All numbers can be divided into prime factors
19:29 < Enigma_Nova> If One is prime, all numbers have an infinite number of factors
19:30 < Enigma_Nova> As you can keep multiplying on One
19:30 < G_Slacker> And the problem with that?
19:30 < Enigma_Nova> Euler's Theorem stuffs up - and so does cryptography - and the internet packet transmission system -
19:30 < Enigma_Nova> 1 is not prime
19:30 < G_Slacker> I can't remember Euler's Theorem right now.
19:31 < Elias> what math are you people taking
19:31 < Enigma_Nova> 3rd year
19:31 < Elias> of what
19:31 < Enigma_Nova> Math
19:31 < G_Slacker> But those are systems that should only count one once, and not as many times as they can divide it out.
19:31 < Enigma_Nova> At university
19:31 < Elias> ??
19:31 < Elias> your 17 right
19:32 < Enigma_Nova> Actually, if you count One once, then you can't count 2 Twice.
19:32 < Enigma_Nova> Factor 4
19:32 < Enigma_Nova> Yes I'm 17
19:32 < Elias> Algebra claclus geomotry what??
19:32 < Enigma_Nova> Start my 3rd year in a few months
19:32 < Elias> 2
19:32 < Elias> 1
19:32 < Enigma_Nova> Got all HDs (A) and Ds (B)
19:32 < G_Slacker> What? How does that follow logically from what I said?
19:32 < G_Slacker> Elias: BC Calc for me.
19:32 < Enigma_Nova> If you can only count One once... then You can't count any other number more than once, by your hypothesis
19:33 < Enigma_Nova> Else why not count one as many times as it divides?
19:33 < Enigma_Nova> Which is of course infinitely many times
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < G_Slacker> One is a special case, thus there are things that should be applied to it that should not be applied to other numbers.
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> 1
19:33 < Elias> etc etc etc
19:33 < Enigma_Nova> 12 = 3*2*2, not 3*2*2*1*1*1*1*1*1*1...
19:34 < G_Slacker> Those two factor sets are equivalent.
19:34 < Enigma_Nova> Primes IIRC are defined that they have no factors except themselves
19:34 < G_Slacker> The only reason you only put one or zero ones is to save reading time.
19:34 < Elias> ahh i think i understand now
19:34 < Enigma_Nova> Of course - They both multiply to 12
19:35 < Elias> enought talk about math this is not school
19:35 < Enigma_Nova> But a prime number cannot have an infinite number of factors.
19:35 < G_Slacker> Elias: Why do you think school subject's shouldn't be discussed outside of school?
19:35 < Enigma_Nova> If you're talking about NON-ONE factors, then 1 falls out of the Fundamental Theorem of Arithmetic as not being prime
19:35 < G_Slacker> Why not?
19:36 < G_Slacker> To the infinite factors statement.
19:36 < Enigma_Nova> A prime number IIRC has a finite number of factors - Namely itself
19:36 < Elias> you spend all day listening about there why do it at home too ?
19:36 < G_Slacker> Here, I do it willingly.
19:36 < Enigma_Nova> I believe the term is modality - 17 has modality 1
19:36 < Elias> actually im just very confused.
19:36 < Enigma_Nova> When all numbers have an infinite number of factors then there is no way of determining which ones are prime
19:37 < G_Slacker> The definition I've always heard for prime numbers is that their only factors are one and themselves.
19:37 < Enigma_Nova> That's a corrupt definition
19:37 < Enigma_Nova> Actually, all mathematicians believe 2 is the first prime
19:37 < Enigma_Nova> (well first positive prime)
19:37 < G_Slacker> You can still determine which ones are prime, it depends on your determination algorithm.
19:38 < Enigma_Nova> Go by Non-one factors, then by the definition of the Fundamental Theory of Arithmetic, One is not prime
19:38 < Enigma_Nova> By definition of a number being the product of its prime factors
19:38 < Elias> maby we should get a 3rd opnion
19:38 < Enigma_Nova> If you disregard One as a factor...
19:38 < Enigma_Nova> .
19:38 < Enigma_Nova> Okay you do that.
19:39 < Enigma_Nova> Find me a Math professor
19:39 < Enigma_Nova> Or a Math Postgraduate
19:39 < Elias> i can e-mail mine
19:39 < Elias> well he is just a high school teacher
19:39 < Enigma_Nova> Sounds good
19:39 < Enigma_Nova> Teachers are Postgraduates.
19:39 < Enigma_Nova> That counts
19:40 < G_Slacker> What you stated as the Fundamental Theory of Arithmetic said nothing to define what makes a number prime.
19:40 < Enigma_Nova> Anyone ever heard of a prime number tree?
19:40 < Enigma_Nova> 12
19:40 < Enigma_Nova> 4 3
19:40 < Enigma_Nova> 2 2 3
19:40 < Elias> I got a 3rd opnion my dad thinks 1 is prime
19:40 < Enigma_Nova> Now if one were prime I could extend that tree down ad infinitum
19:40 < Enigma_Nova> Well your dad is wrong
19:41 < G_Slacker> And what is your problem with that tree branching infinitely?
19:41 < Enigma_Nova> 1 is not officially prime
19:41 < Aidun> I'm going offline, catching sleep in the left hours
19:41 < Enigma_Nova> I'm off to see a mathematical website to bloody SOURCE my validity...
19:41 < G_Slacker> What is the reason for this perverse avoidance of infinity?
19:43 < Enigma_Nova> So the set is Countable and Finite
19:43 < Enigma_Nova> Stuff is easier to work with if you define it like that
19:44 < Enigma_Nova> http://teachers.net/mentors/math/top....14.45.17.html
19:44 < Enigma_Nova> See it for yourselves
19:45 < Enigma_Nova> http://teachers.net/mentors/math/top....22.08.38.html
19:45 < Enigma_Nova> That one mentions the Fundamental Theorem of Arithmetic
19:45 < Enigma_Nova> Just what I've been going on about
19:46 -!- Aidun [Aidun@xs195-241-31-18.dial.12move.nl] has quit [Quit: sleeping, again thanks to Chieftess!]
19:47 < G_Slacker> I talked to my brother, who went rather far into math, and he says that most college level books will say 1 is prime, and most research mathematticians make the decision based on how it affects the problem they are working on.
19:47 < G_Slacker> Reading links now.
19:48 < Enigma_Nova> Fundamental Theorem of Arithmetic fails if One is considered prime...
19:48 < Enigma_Nova> If the problem you are working on negates the FTA then so be it
19:48 < Enigma_Nova> But I wouldn't deny myself such a useful theory
19:48 < Elias> the only use for knowing if a number is prime is so you can tell some one else it is prime
19:48 < Enigma_Nova> ... Actually 1 is neither Prime nor Composite.
19:49 < Enigma_Nova> You ignorant fool! Primes are used quite heavily in Cryptography
19:49 < Elias> 1 and 0 always have execptions
19:49 < Enigma_Nova> Bloody 0
19:49 < Elias> is 0 prime
19:49 < Elias> no
19:49 < Enigma_Nova> x / x = 1 Unless x = 0
19:49 < Elias> is it composite
19:49 < G_Slacker> "there would be no
19:49 < G_Slacker> unique prime factorization"
19:50 < Enigma_Nova> 0 is neither prime nor composite...
19:50 < G_Slacker> Garbage.
19:50 < G_Slacker> To what I quoted.
19:50 < Enigma_Nova> Let me prove it.
19:50 < Enigma_Nova> 2 = 2
19:50 < Enigma_Nova> 2 = 2 * 1
19:50 < Elias> why is 0 considerd positive?
19:50 < Enigma_Nova> There. different.
19:50 < Enigma_Nova> It isnt
19:50 < G_Slacker> 0 is neither positive or negative.
19:50 < Enigma_Nova> It's that midway point
19:51 < G_Slacker> But some braindead teachers try to claim it's positive.
19:51 < Enigma_Nova> Prolly the same ones that violate the FTA
19:51 < Enigma_Nova> ...I wonder what they do about negative numbers...
19:51 < G_Slacker> Enigma_Nova: How is that factorization not unique?
19:51 < Enigma_Nova> The set containing: 2
19:51 < Enigma_Nova> Differs from the set containing: 2, 1
19:52 < Enigma_Nova> *sigh* If I have a red marble and a green marble... that's not the same as having only a red marble, even if the green marble has no value
19:52 < Enigma_Nova> 2 = 2, and 2 = 2*1
19:52 < Enigma_Nova> Both are factorisations
19:52 < G_Slacker> Both of which are incomplete if 1 is considered prime.
19:52 < Enigma_Nova> But only one is the PRIME factorisations.
19:53 < Enigma_Nova> If 1 is considered prime than any number of multiplications can be done.
19:53 < G_Slacker> To your analogy: You simply aren't counting the things you have 0 of for convenience.
19:53 < Enigma_Nova> Quite true
19:54 < Enigma_Nova> 12 = 2^2 * 3^1 * 5^0 * 7^0 * ... 1^Anything.
19:54 < Elias> 4th and 5th opnion my college bro thinks its prime but my twin bro thinks its not
19:54 < G_Slacker> Thus, the only reason 1 could be considered non prime is purely for matters of convenience.
19:54 < Enigma_Nova> Or why it would be would be for convenience
19:54 < Enigma_Nova> Namely, the FTA
19:54 < G_Slacker> Leaving out matters of convenience, 1 is prime.
19:55 < Elias> I agree with G
19:55 < Enigma_Nova> Ignore an important mathematical theorem and it becomes true
19:55 < Enigma_Nova> Sure... ignore what makes it false and everything becomes true
19:55 * Enigma_Nova is darn well posting this in the OT forum
19:55 < Elias> lol
19:55 < Elias> im gonna have to see that
19:56 < G_Slacker> I'm still looking for an exact stating of the FTA.
19:56 < Enigma_Nova> The Great Prime Debate

[geeslaka]

Thanks Gepap.

[Inverse Icarus]

one is prime.

the simplest definition of prime is any number that is only divisible by itself and one (note, 2 factors, not infinate amount of ones), and one fits into that description.

btw, funny:

19:31 < Elias> what math are you people taking
19:31 < Enigma_Nova> 3rd year
19:31 < Elias> of what
19:31 < Enigma_Nova> Math

[GePap]

Just thanks to the chatlog it should win without question.

[geeslaka]

Found a statement of the FTA in Enigma_Nova's link.

The
FDA states that every positive integer greater than one can be
expressed uniquely as a product of primes, apart from the
rearrangement of terms.

http://teachers.net/mentors/math/top....10.03.55.html

One being prime does not void the FTA because all the factorisations stated as being different but for the same number are incomplete. Any factorisation that does not include 1^infinity is incomplete.

[Enigma_Nova]

The FTA mentions nothing about complete factorisations.
It mentions unique prime factorisations.
Which if 1 is prime would make these not unique.

A factorisation is a factorisation if it multiplies to give the desired number.

[SnowFire]

1^infinity is whatever the heck you want it to be. So that seems dangerous to stick in a factorization.

Anyway, as noted before, having unique factorizations is a Good Thing, among other reasons.

[geeslaka]

The FTA mentions nothing about complete factorisations.
It mentions unique prime factorisations.
Which if 1 is prime would make these not unique.

To determine uniqueness, the whole must be considered. If the whole is not considered, then there is still doubt about the uniqueness. The factorisations cited as proof are not whole, and thus cannot be compared for uniqueness.

A factorisation is a factorisation if it multiplies to give the desired number.

True.

[geeslaka]

1^infinity is whatever the heck you want it to be. So that seems dangerous to stick in a factorization.

When evaluated, 1^infinity is not "whatever the heck you want it to be", it is 1.

Anyway, as noted before, having unique factorizations is a Good Thing, among other reasons.

1 being prime does not eliminate unique factorizations.

[Urban Ranger]

Originally posted by geeslaka
When evaluated, 1^infinity is not "whatever the heck you want it to be", it is 1.

Therefore, 1^Infinity is meaningless.

[geeslaka]

Therefore, 1^Infinity is meaningless.

It is not meaningless, it is a statement that the complete factorisation is being used.

[Urban Ranger]

Originally posted by geeslaka

Therefore, 1^Infinity is meaningless.

It is not meaningless, it is a statement that the complete factorisation is being used.

Of course it is, because 1^Infinity evaluates to 1, nothing more.

Do not forget 1 is unity in multiplication, therefore it can be safely removed from all multiplications and divisions. Of course, you can also introduce terms that evaluates to 1, and that's one of the tricks in algebra.

[geeslaka]

Originally posted by Urban Ranger
Of course it is, because 1^Infinity evaluates to 1, nothing more.

Factorisation is the process of finding the complete set of numbers that when multiplied together, will evaluate to the number being factored.
Thus, evaluating part of the factorisation defeats the purpose.

Do not forget 1 is unity in multiplication, therefore it can be safely removed from all multiplications and divisions. Of course, you can also introduce terms that evaluates to 1, and that's one of the tricks in algebra.

]
Exactly my point, the only reason 1^infinity is not included everywhere, is convenience. When it's needed, it's put in, when it's not, it's taken out.

Edit:
As I stated before:

19:54 < G_Slacker> Thus, the only reason 1 could be considered non prime is purely for matters of convenience.
19:54 < Enigma_Nova> Or why it would be would be for convenience
19:54 < Enigma_Nova> Namely, the FTA
19:54 < G_Slacker> Leaving out matters of convenience, 1 is prime.

[Urban Ranger]

Originally posted by geeslaka

Originally posted by Urban Ranger
Of course it is, because 1^Infinity evaluates to 1, nothing more.

Factorisation is the process of finding the complete set of numbers that when multiplied together, will evaluate to the number being factored.
Thus, evaluating part of the factorisation defeats the purpose.

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

[geeslaka]

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

The non-trivial part is there purely for convenience.

[connorkimbro]

the monkey says, 1 is prime! and you know the monkey must be smart, because he's wearing glasses.

[geeslaka]

Are you sure that's a monkey? I've seen old guys that look like that. Specifically my grand father.

[Jaguar]

[Urban Ranger]

Originally posted by geeslaka

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

The non-trivial part is there purely for convenience.

You disregard the non-trivial part because adding it in makes absolutely no difference to the end result.

By your logic, any additions and subtractions cannot be considered complete since you get to play with an infinite number of 0's.

[geeslaka]

You disregard the non-trivial part because adding it in makes absolutely no difference to the end result.

True in some cases, but it does make a difference when determining uniqueness of prime factorisations.

By your logic, any additions and subtractions cannot be considered complete since you get to play with an infinite number of 0's.

True. What is wrong with this ability?

[Kuciwalker]

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

[geeslaka]

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

Name a few of those properties. And this definition is still a matter of convenience.

[Boris Godunov]

Originally posted by geeslaka
Found a statement of the FTA in Enigma_Nova's link.

The
FDA states that every positive integer greater than one can be
expressed uniquely as a product of primes, apart from the
rearrangement of terms.

I think it's great the Food and Drug Administration is finally starting to get a handle on these mathematical issues.

[geeslaka]

I simply ignored that typo as not worthy of notice.

[SnowFire]

Wrong wrong wrong, folks. 1^infinity might be 1. It might not be. It's indeterminate, much like 0/0. The 0/0 case is obvious: 6/2 is cleary 3 because you need to add 3 2's together to get 6. How many 0's do you have to add together to get 0? Well, if you don't add any (0), that works. But if you add 5 0's, you get 0 too. In fact, if you add negative 10 billion 0's, you also get 0. This is why when you get 0/0 as the limit of some function, you have no clue what the answer is and need to use more advanced techniques. Outside of a limit, 0/0 is simply indeterminate.

It may be harder to understand why 1^infinity is the same. But basically, when you multiply a number infinite times, things get wonky. Don't forget that a number infinitely close to 1 is still "1" (just as .9999... = 1), but if you do infinite multiplications on it, you might be able to "undo" that closeness and bring out some larger number.

Here's a webpage I found with some more examples:
http://www.sosmath.com/calculus/indf...tro/intro.html

So 1^infinity is not a good option for prime factorizations, since that "factorization" might not multiply to the number we think it does.

[Kuciwalker]

Originally posted by geeslaka

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

Name a few of those properties. And this definition is still a matter of convenience.

Why is the number one not a prime?

The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below are four answers, each more technical than its precursor.

[b]Answer One: By definition of prime[bb]

The definition is as follows:

[i]An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.[i]

Clearly one is left out, but this does not really address the question "why?"

Answer Two: Because of the purpose of primes.

The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved:

The Fundamental Theorem of Arithmetic

Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size.

Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1.a = 1.1.a = ... That is, divisibility by one fails to provide us any information about a

Answer Three: Because one is a unit.

Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units.
So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with with the number one (and with primes).

Answer Four: By the Generalized Definition of Prime.

There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they mustbe found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:"

An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D.

Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.

[geeslaka]

I cede to the more advanced mathematical knowledge.
SnowFire is correct because infinity does not have to be a whole number and I was not considering that case.

Thank you skywalker.

[Kuciwalker]

You're welcome. Google is useful, n'est-ce pas?