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Old 04-14-2019, 02:57 PM
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Why is 1 not a prime number?


According to Wikipedia, "1 is not prime, as it is specifically excluded in the definition." Why is 1 excluded? What's the reasoning for not including it? Does it lead to some sort of contradiction that I'm not seeing?

Thanks in advance for your answers.
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Old 04-14-2019, 03:08 PM
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For most mathematical facts that apply to prime numbers, it's simpler if 1 isn't one of them. Most notably is the Unique Prime Factorization Theorem: Any positive integer can be expressed as the product of primes in exactly one way. For instance, 42 = 2*3*7. But if we allowed 1 as a prime, then we could also say that 42 = 1*2*3*7, or 42 = 1*1*1*1*2*3*7, or whatever.
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Old 04-14-2019, 06:06 PM
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Interesting article on the subject:

https://blogs.scientificamerican.com...-prime-number/
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Old 04-14-2019, 10:44 PM
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It's a matter of convenience. Historically, 1 was often considered prime. But if 1 is prime, then it turns out there's a lot of stuff where you have to say "for all prime numbers, except 1" to get it to work. So a consensus evolved that the definition of prime numbers would exclude 1.
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Old 04-14-2019, 11:19 PM
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Primes (excluding 1) are like single-character strings, of length 1; "a", "b", "c", "d".

1 is like the empty string, of length 0; "".

Composite numbers are like strings of length > 1; "hi", "jack", "afkafshaf", etc. They're made up of multiple single-character strings stuck together. Every string decomposes in an essentially unique way as a composition of single characters.

The single-character strings have much, much more in common with each other than they do with the empty string. And so it is convenient to have a nice name for them. (In string-world, we call them "characters" or "letters". In the multiplying-numbers-world side of this analogy, we call them "prime numbers").

It's true that if you consider the property "My only substrings are myself and the empty string" (i.e., "I am a string of size ≤ 1"), it is satisfied by both the single-letter strings and the empty string satisfy. Sometimes one might care to consider this property; it might as well be given a name too, if you find yourself using it a lot. But much less often does this concept naturally come up.

Last edited by Indistinguishable; 04-14-2019 at 11:23 PM.
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Old 04-15-2019, 02:40 AM
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Composite numbers are like strings of length > 1; "hi", "jack", "afkafshaf", etc. They're made up of multiple single-character strings stuck together. Every string decomposes in an essentially unique way as a composition of single characters.
True, but only if you define string concatenation to commute, such that "a" : "b" is considered to be the same string as "b" : "a", and this doesn't usually work in most contexts where strings are being concatenated.
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Old 04-15-2019, 03:15 AM
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The inequality li(x) > π(x) holds for all small numbers, and perhaps for all positive integers less than 10^316. Nobody has ever demonstrated a specific exception to li(x) > π(x), although it is now known that there are many googols of such exceptions near 10^316.

Even ignoring x=1, if the definition of prime is changed to include 1, then 2, 3 and 4 would all be exceptions to li(x) > π(x).
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Old 04-15-2019, 05:51 AM
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Are -2, -3, -5, ... primes? I’m guessing not because they would be divisible by -1 in addition to 1 and themselves.
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Old 04-15-2019, 07:28 AM
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Are -2, -3, -5, ... primes? Im guessing not because they would be divisible by -1 in addition to 1 and themselves.
If you count negative integers as numbers, then you have to take them into account and yes, they are; units like 1 and -1 are not counted in the factorization.

It might also be simpler to think of a prime number as one for which whenever p divides ab, then p divides a or p divides b (and p is not zero or a unit).
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Old 04-15-2019, 07:51 AM
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In school we were taught that one is not a prime number based on this definition of a prime number: "A prime number is a whole number greater than 1 whose only factors are 1 and itself."

The fact that one IS itself makes it suspiciously odd even to a grade school student. LOL
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Old 04-15-2019, 08:28 AM
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Because it is a unit and units are not prime.
This is also why -1 is not prime in the integers and there are no primes in the rationals
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Old 04-15-2019, 11:32 AM
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Since it is a matter of definition, we define prime in the most convenient way possible. And that is to exclude 1 (and -1) for all the reasons mentioned above. It is just simpler that way. The primes are the atoms of the numbers and we don't count nothing as an atom.

Incidentally, if you look at the Gaussian integers, numbers of the form a + bi, where a and b are integers and i is the square root of -1, you now have to exclude 1, -1, i, and -i from being counted as primes. Also 2 = (1+i)(1-i) is no longer prime. Nor is 5 = (2+i)(2-i). But 1+i, 2+i, and 3 are all prime.
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Old 04-15-2019, 12:44 PM
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True, but only if you define string concatenation to commute, such that "a" : "b" is considered to be the same string as "b" : "a", and this doesn't usually work in most contexts where strings are being concatenated.
Sure; I'm not saying it's an exact isomorphism (although it is exact up to commutativity), just that this is an analogy to another example one might think about where the same phenomenon is in play, of why "things of simplicity level exactly 1" tends to be a more natural concept than "things of simplicity level either 0 or 1".

The prime numbers all have far more in common with each other than they do with 1, even though both are "simpler" than composite numbers, in just the same way that the single letters all have far more in common with each other than they do with the empty string, even though both are "simpler" than longer strings.

And this same kind of phenomenon arises over and over elsewhere in mathematics as well (see https://ncatlab.org/nlab/show/too+simple+to+be+simple, though this will not be a useful page for anyone who is not already a certain kind of mathematician).

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Old 04-15-2019, 12:54 PM
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It might also be simpler to think of a prime number as one for which whenever p divides ab, then p divides a or p divides b (and p is not zero or a unit).
The parenthetical exclusion of units will seem ad hoc, but should be unified with the rest of the definition:

A prime number, in this sense, is one which only divides the product of some list of values when there is some value in that list which it divides. This is true not just of lists of 2 values, but also lists of 3 values, lists of 4 values... and even lists of 0 values.

A prime only divides the product of an empty list (which comes out to 1) when there is some value in that list which it divides (which of course can't happen, since the list is empty); thus, a prime is not allowed to divide 1.

So the exclusion of 1 and other units from the prime numbers in this sense is hardly ad hoc; it's just part of the same divisibility condition. (The exclusion of zero from the primes is genuinely a separate condition here, though; it would otherwise be a, well, prime example of such a thing.)
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Old 04-15-2019, 01:38 PM
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Guy who asked about negative numbers here ...

So most properly, the list of primes is ...-5, -3, -2, 2, 3, 5 ..., eh?

If I ever found the largest prime, I would take its negative and say I found the smallest prime, just to see how the popular media would deal with that!
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Old 04-15-2019, 03:02 PM
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Guy who asked about negative numbers here ...

So most properly, the list of primes is ...-5, -3, -2, 2, 3, 5 ..., eh?
It depends on the context.

Most commonly, when people talk about prime numbers, they're working with the set of natural numbers (positive integers), in which the primes would only be positive 2, 3, 5, .... You wouldn't include -2, -3, -5, ... because they don't exist within that context. In other contexts, they do exist, but they might or might not be prime depending on what other numbers exist.
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Old 04-15-2019, 03:35 PM
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The parenthetical exclusion of units will seem ad hoc, but should be unified with the rest of the definition:

A prime number, in this sense, is one which only divides the product of some list of values when there is some value in that list which it divides. This is true not just of lists of 2 values, but also lists of 3 values, lists of 4 values... and even lists of 0 values.

A prime only divides the product of an empty list (which comes out to 1) when there is some value in that list which it divides (which of course can't happen, since the list is empty); thus, a prime is not allowed to divide 1.

So the exclusion of 1 and other units from the prime numbers in this sense is hardly ad hoc; it's just part of the same divisibility condition. (The exclusion of zero from the primes is genuinely a separate condition here, though; it would otherwise be a, well, prime example of such a thing.)
Since this might be clearer with some examples: Suppose that I want to know if 6 is prime. I can compare it to the list {4,7,9}, for instance. 6 does not divide any of those numbers... but if I multiply them all together, I get 252, and 6 does divide 252 (252 = 6*42). So since there is a list of numbers that 6 doesn't divide, but it does divide their product, 6 is composite.

By contrast, suppose I try that with 7. I could try the list {1,10,11,14}, and the product of that list is 1540, and 7 divides that (1540 = 7*220)... but 7 also divides 14. Alternately, I could use {5,12,13,27,242} as my list, and 7 doesn't divide any of those... but neither does it divide the product. No matter what list I try, either 7 doesn't divide the product, or it does divide one of the numbers on the list. So 7 is prime.

Now we try it with 1: No matter what (non-empty) list of numbers I come up with, 1 always divides all of the numbers on that list. So it's not prime, but it's not-prime in a different way than 6 is not-prime. It's a unit.

Or 0, of course: No matter what list of numbers I come up with, 0 never divides the list's product, because 0 doesn't divide anything. It's yet another kind of non-prime, but there's no special name for that kind of non-prime, because 0 is the only one.

And this definition also works when we expand our scope to include negative integers, or complex integers. I suppose that you could even extend it to the rationals or reals, but that's kind of boring, because in those contexts, everything (except 0) is a unit.
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Old 04-15-2019, 04:41 PM
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The parenthetical exclusion of units will seem ad hoc, but should be unified with the rest of the definition:

A prime number, in this sense, is one which only divides the product of some list of values when there is some value in that list which it divides. This is true not just of lists of 2 values, but also lists of 3 values, lists of 4 values... and even lists of 0 values.

A prime only divides the product of an empty list (which comes out to 1) when there is some value in that list which it divides (which of course can't happen, since the list is empty); thus, a prime is not allowed to divide 1.

So the exclusion of 1 and other units from the prime numbers in this sense is hardly ad hoc; it's just part of the same divisibility condition. (The exclusion of zero from the primes is genuinely a separate condition here, though; it would otherwise be a, well, prime example of such a thing.)
It's a nice, clean definition, but I have no doubt that what actually happened was, as freido (and Chronos) said, mathematicians realized they talked about {prime numbers not including one} far more than {prime numbers including one} and so agreed to define "prime numbers" to exclude one, to save having to say "except one" all the time when they talked primes. Only after that agreement was there a reason to find a definition of prime that naturally excludes one (after all, it must be admitted that that definition is much less clear and intuitive than the usual 'no factors except itself and one').
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Old 04-15-2019, 04:57 PM
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If negative numbers are included in the definition of prime numbers you no longer have unique prime factorization. 35 will be both 5*7 and -5*-7 for instance.
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Old 04-15-2019, 05:51 PM
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If negative numbers are included in the definition of prime numbers you no longer have unique prime factorization. 35 will be both 5*7 and -5*-7 for instance.
This was glossed above; you need to be precise in that case what you mean by "unique factorization". If p equals q times a unit, so that p divides q and also q divides p, then p and q are not really different primes. So, in your example, 5*7 and -5*-7 are not truly different factorizations.

Now for a counterexample you can try adjoining √-5 to the integers. Then you really do lose unique factorization, since, e.g., 6 = 2*3 = (1+√-5)(1-√-5).
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Old 04-15-2019, 07:04 PM
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That only loses you prime factorization if 1+sqrt(-5) is a prime. But it divides 2*3, even though it doesn't divide 2 or 3, so it's composite, and (1+sqrt(-5))(1-sqrt(-5)) is not a prime factorization.

Although... then I suppose you can ask what the prime factorization of sqrt(-5) is.

But on the gripping hand, just adjoining a single complex irrational to the integers breaks all sorts of properties, like closure.
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Old 04-15-2019, 08:38 PM
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That only loses you prime factorization if 1+sqrt(-5) is a prime. But it divides 2*3, even though it doesn't divide 2 or 3, so it's composite, and (1+sqrt(-5))(1-sqrt(-5)) is not a prime factorization.

Although... then I suppose you can ask what the prime factorization of sqrt(-5) is.

But on the gripping hand, just adjoining a single complex irrational to the integers breaks all sorts of properties, like closure.
Sorry, you are right. Adjoining sqrt(-5) breaks prime factorization because, as the example shows, not every number factors into primes in the first place. Once every element factors into primes, you have uniqueness in the usual way. Another way to look at the example would be that the fact that there are two essentially different factorizations into irreducible elements shows that the ring in question fails to be a unique factorization domain.

What property of "closure" did you mean is broken, though?

ETA isn't sqrt(-5) itself already prime?

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Old 04-15-2019, 10:20 PM
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I've not ever seen a definition of prime numbers that allows for negative numbers. They all say "an integer [or whole number] greater than 1..." or "can only be evenly divided by itself and 1."

Can anyone show me a cite for the idea that negative numbers can also be considered prime?
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Old 04-15-2019, 10:36 PM
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I've not ever seen a definition of prime numbers that allows for negative numbers. They all say "an integer [or whole number] greater than 1..." or "can only be evenly divided by itself and 1."

Can anyone show me a cite for the idea that negative numbers can also be considered prime?
Algebraic Number Theory, by J. S. Milne, Introduction, page 1:
Quote:
The fundamental theorem of arithmetic says that every nonzero integer m can be written in the form,

 m = p1...pn,   pi a prime number,

and that this factorization is essentially unique.

Consider more generally an integral domain A. An element a ∈ A is said to be a unit if it has an inverse in A (element b such that ab = 1 = ba). I write A for the multiplicative group of units in A. An element π of A is said to [be] prime if it is neither zero nor a unit, and if

 π | ab ⇒ π | a or π | b .

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Old 04-15-2019, 10:49 PM
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NB I see that he explains more clearly what I said in my reply to Chronos :

"Why does unique factorization fail [in Z[√-5]]? The problem is that irreducible elements need not be prime. In the above example, 1 + √-5 divides 2⋅3 but it divides neither 2 nor 3. In fact, in an integral domain in which factorizations exist... factorization is unique if all irreducible elements are prime."
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Old 04-16-2019, 08:09 AM
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Algebraic Number Theory, by J. S. Milne, Introduction, page 1:
That just shows how you can define a prime factorization of a negative number though, and elsewhere in the same work he writes:
Quote:
Throughout the notes, p is a prime number, i.e.,
p = 2, 3, 5, ...
Not saying you're wrong, as I'm out of my depth here. But that specific quote, and the notes on basic definitions used in the work, seem to stick with only positive primes.
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Old 04-16-2019, 08:41 AM
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That just shows how you can define a prime factorization of a negative number though
It looks that way to me, too. The allows the m to be negative even though the p's are not.
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Old 04-16-2019, 09:18 AM
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I quoted some context, but the last sentence defines what it means for an element of an integral domain to be "prime", namely that if p divides ab then it divides one of the factors a or b. This is a generalization of what it means for a positive rational integer to be prime (the rational integers being the prototypical example of an integral domain). In particular, negative integers are "prime" under this definition, and multiplying a "prime" by a unit leaves it prime.

Let us ask, can anyone cite a book or article wherein negative numbers are not prime, as soon as negative integers are considered, rather than only looking at natural numbers? Because the text I linked to is not the only one where negative numbers may be prime.
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Old 04-16-2019, 09:51 AM
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I quoted some context, but the last sentence defines what it means for an element of an integral domain to be "prime", namely that if p divides ab then it divides one of the factors a or b. This is a generalization of what it means for a positive rational integer to be prime (the rational integers being the prototypical example of an integral domain). In particular, negative integers are "prime" under this definition, and multiplying a "prime" by a unit leaves it prime.

Let us ask, can anyone cite a book or article wherein negative numbers are not prime, as soon as negative integers are considered, rather than only looking at natural numbers? Because the text I linked to is not the only one where negative numbers may be prime.
The book you quote states : "Throughout the notes, p is a prime number, i.e., p = 2, 3, 5, ..."

In a very practical sense that means that -2 might be "prime", but it's the prime 2 with a negative unit.
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Old 04-16-2019, 02:16 PM
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Nobody talks much about the primes in Z, because the negative primes are just the familiar primes in N with a negative sign, so there's nothing new to be said. On the other hand, the primes in the complex integers are different, as evidenced by 5 not being prime in that domain, so they get attention again.
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Old 04-16-2019, 04:23 PM
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Disclaimer: I really don't know anything about number theory, especially analytic number theory.

Still, may I ask: do primes in other domains also link to the Riemann Hypothesis? Or are there separate analogues? Neither?
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Old 04-17-2019, 01:45 PM
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Disclaimer: I really don't know anything about number theory, especially analytic number theory.

Still, may I ask: do primes in other domains also link to the Riemann Hypothesis? Or are there separate analogues? Neither?
I read this as a two-part question.

One can define an analogue of the classical Riemann zeta function for any algebraic number field; this was done by Dedekind in 1863, and today one speaks of Dedekind zeta-functions. (And one can obtain results on the distribution of primes in number fields.) This naturally leads to an "extended" Riemann hypothesis which would apply to any number field. (And of course such zeta functions may be and are generalized even further.) Note that the various generalized Riemann hypotheses do include as a particular case the classical Riemann hypothesis.

Now, the first part of your question seems to be whether the ordinary RH can be reformulated as an equivalent problem involving the distribution of primes in a more general setting, more generalized zeta functions or some other arithmetic or analytic problem. Certainly mathematicians have endeavored to establish such bridges, because you need non-elementary tools with which to attack the problem, but I don't necessarily feel qualified to survey all the precise statements. It is worth pointing out that today no one yet knows a proof of the RH so it's not like there is one obvious thing to try.
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Old 04-17-2019, 06:22 PM
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All of which indeed proves, once and for all:

One *is* the loneliest number. QED.
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Old 04-17-2019, 06:26 PM
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Nah, zero is the loneliest number. 1 has -1, i, and -i to keep it company in the Units Club, but there's only one zero.
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Old 04-17-2019, 06:36 PM
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I read this as a two-part question.

One can define an analogue of the classical Riemann zeta function for any algebraic number field; this was done by Dedekind in 1863, and today one speaks of Dedekind zeta-functions. (And one can obtain results on the distribution of primes in number fields.) This naturally leads to an "extended" Riemann hypothesis which would apply to any number field. (And of course such zeta functions may be and are generalized even further.) Note that the various generalized Riemann hypotheses do include as a particular case the classical Riemann hypothesis.

Now, the first part of your question seems to be whether the ordinary RH can be reformulated as an equivalent problem involving the distribution of primes in a more general setting, more generalized zeta functions or some other arithmetic or analytic problem. Certainly mathematicians have endeavored to establish such bridges, because you need non-elementary tools with which to attack the problem, but I don't necessarily feel qualified to survey all the precise statements. It is worth pointing out that today no one yet knows a proof of the RH so it's not like there is one obvious thing to try.
Thank you for answer. Very helpful. What you note is very much the type of thing I was wondering about.
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Old 04-18-2019, 07:04 AM
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Nah, zero is the loneliest number. 1 has -1, i, and -i to keep it company in the Units Club, but there's only one zero.
Not in two's complement binary. 00...0000 and 10...000 are both zeros. The first is positive zero and the second is negative zero. They both have the property that if you negate them in th usual way (flip all bits and add one) you get the same number back.

So, the negation of positive zero is positive zero and the negation of negative zero is negative zero.

Somehow that sentence makes sense to us computer folk.

(Note that since "-0" isn't often be used in practice, some people define it as the negative of 2 raised to the (word size - 1) in order to "squeeze out" an extra value. This is an ugly, ugly kludge that breaks a lot of binary arithmetic and if you need that extra value you should be using a larger word size.)
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Old 04-18-2019, 07:59 AM
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The parenthetical exclusion of units will seem ad hoc, but should be unified with the rest of the definition:

A prime number, in this sense, is one which only divides the product of some list of values when there is some value in that list which it divides. This is true not just of lists of 2 values, but also lists of 3 values, lists of 4 values... and even lists of 0 values.

A prime only divides the product of an empty list (which comes out to 1) when there is some value in that list which it divides (which of course can't happen, since the list is empty); thus, a prime is not allowed to divide 1.

So the exclusion of 1 and other units from the prime numbers in this sense is hardly ad hoc; it's just part of the same divisibility condition. (The exclusion of zero from the primes is genuinely a separate condition here, though; it would otherwise be a, well, prime example of such a thing.)
Quoting this post just to say "welcome back" (I haven't seen you post in a while, though that could just be me) and to note that rarest of beasts - an unedited post by Indistinguishable .
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Old 04-18-2019, 10:29 AM
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Not in two's complement binary. 00...0000 and 10...000 are both zeros. The first is positive zero and the second is negative zero. They both have the property that if you negate them in th usual way (flip all bits and add one) you get the same number back.
That's just a representation of numbers though, we can introduce -0 in all the other representations as well, it just wouldn't give us anything useful.
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Old 04-18-2019, 10:38 AM
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That's just a representation of numbers though, we can introduce -0 in all the other representations as well, it just wouldn't give us anything useful.
But it would encourage some people to ask about +0, and then there would have to be a whole argument on the subtle implications their differences. Mathematicians don't need any more toys.

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Think of a number. Now try not to.
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Old 04-18-2019, 11:06 AM
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In mathematics, +0 = -0. The computer engineer's nightmare therefore begins as soon as these have different or multiple machine-level representations, because the hardware will have to check for it every time there is an equality test or comparison, and who knows what bugs could still ensue.

Back to mathematics, in a ring (i.e. algebraic structure in which you have addition and multiplication) you always have to have zero, but it is conceivable that 1 = 0, in which case zero could claim that I am the Alpha and the Omega, the First and the Last, the Beginning and the End.
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Old 04-18-2019, 01:13 PM
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Originally Posted by ftg View Post
Not in two's complement binary. 00...0000 and 10...000 are both zeros. The first is positive zero and the second is negative zero. They both have the property that if you negate them in th usual way (flip all bits and add one) you get the same number back.
One's complement. One of the advantages of two's complement is that it lacks a negative zero, which is a complication you don't need in integer arithmetic.

(It does have some utility in floating point, where it can represent an underflow and the sign thereof.)

Last edited by Derleth; 04-18-2019 at 01:13 PM.
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Old 04-18-2019, 01:16 PM
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Nah, zero is the loneliest number. 1 has -1, i, and -i to keep it company in the Units Club, but there's only one zero.
And there are indeed an infinite number of roots of unity, all of which are just points on the unit circle in the complex plane.
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Old 04-18-2019, 01:19 PM
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Originally Posted by DPRK View Post
Back to mathematics, in a ring (i.e. algebraic structure in which you have addition and multiplication) you always have to have zero, but it is conceivable that 1 = 0, in which case zero could claim that I am the Alpha and the Omega, the First and the Last, the Beginning and the End.
0 is already a ring.

~Max
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Old 04-18-2019, 04:50 PM
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Yeah, but what DPRK is saying is that {0} is also a ring.
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Old 04-20-2019, 05:25 AM
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Quoting this post just to say "welcome back" (I haven't seen you post in a while, though that could just be me) and to note that rarest of beasts - an unedited post by Indistinguishable .
Thanks! Although, back into my reduced posting hole I likely go, perhaps to pop up next again in several months, who knows? (Apparently, my last post before this thread was a one minor note in August 2018, and before that in March 2018)

Last edited by Indistinguishable; 04-20-2019 at 05:27 AM. Reason: I have no great plan. The ebbs and flows of interest.
  #46  
Old 04-20-2019, 08:27 AM
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And, yup, there's an edit on that post, thereby restoring the Cosmic Indistinguishable Balance.
  #47  
Old 04-20-2019, 09:09 AM
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Let me get back to the zero ring {0}. It is a ring, actually called the terminal ring, the one that every ring has a unique homomorphism to. I recall one book, I have forgotten which, in which 1 != 0 is part of the definition. That's reasonable for fields where you don't expect a terminal object, but for rings is simply wrong and anathema to category theorists like me.

The simplest field has two elements 0 and 1 with 1 + 1 =0.
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Old 04-20-2019, 01:29 PM
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I suppose that it depends on why you're talking about rings. If you're viewing a ring as a context, and studying the relationships between different elements of a ring, then {0} is pretty useless, because there's nothing interesting to say within that context. But if rings themselves, and the relationships between them, are what you're studying, then the trivial ring {0} is probably just as important as the trivial number 1 or 0.
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Old 04-20-2019, 02:32 PM
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We were talking about definitions that let us make natural statements cleanly without corner cases before, and in that same vein, expanding on what Hari said, if one rules out the terminal ring as not really a ring, all kinds of statements suddenly acquire corner cases; e.g., "a ring modulo any set of elements is a ring... UNLESS 1 can be written as a sum of left-multiples of right-multiples of values in that set".

There is the fairly natural notion stronger than a ring of an "integral domain", though, which is a ring which can be embedded into a field. These are the same as commutative rings where 0 has the primality property that no product of nonzero values is zero (including the empty case that 1 isn't 0).

People sometimes misdefine integral domains ignoring the empty case and calling the terminal one element ring an integral domain. This inclusion is just as much a wart as it's exclusion from rings would be!

Last edited by Indistinguishable; 04-20-2019 at 02:36 PM.
  #50  
Old 04-20-2019, 03:42 PM
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Regarding primes, if R is a ring (let's assume commutative, and containing 1), and like number theorists we consider prime ideals rather than just prime numbers, then what is supposed to happen is that P is prime if and only if R modulo P is an integral domain. The example of the integers shows that (0) is a prime ideal in the ring of integers, for Z = Z/(0) is an integral domain. (Even though 0 is not normally counted as a prime number.) But, similar to the subject of this thread, (1) is not, and Z/(1) = the terminal one-element ring should not be an integral domain!

Relatedly, when people talk about "the field with one element", the thing to keep in mind is that this is just a manner of speaking because there isn't one. As reiterated above the smallest field has 2 elements, namely 0 and 1.
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