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?
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:
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.”
That just shows how you can define a prime factorization of a negative number though, and elsewhere in the same work he writes:
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.
It looks that way to me, too. The ± allows the m to be negative even though the p’s are not.
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.
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.
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.
All of which indeed proves, once and for all:
One is the loneliest number. QED.
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.
Thank you for answer. Very helpful. What you note is very much the type of thing I was wondering about.
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.)
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 :).
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.
Tris
Think of a number. Now try not to.
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.