-1 as a prime
james dolan
>-1, factors: -1, 1. HEY! NEGATIVE
>ONE IS PRIME! I HEREBY PATENT MY DISCOVERY OF THE WORLD'S
>FIRST NEGATIVE PRIME NUMBER!11!!
sorry, you'll have to get in line behind all of the other cranks and
kooks before you who've made the same "discovery". just last week i
was reading a book where the author presents the idea that -1 is a
prime as his own brilliant idea. he gives a list of four example
statements that he claims are stated most concisely if -1 is
considered to be a prime. his first example is "every nonzero
rational number is uniquely a product of powers of prime numbers p";
apparently he's too stupid to realize that -6 is both 2*3*(-1) and
2*3*(-1)*(-1)*(-1). his second example is "for distinct odd primes
f(p,q) and f(q,p) differ just when p = q = -1 (mod 4)", where f is the
"legendre symbol" function. this is just too brilliant for words:
he's trying to argue that -1 should be treated on the same footing as
the official genuine primes, and he does so by exhibiting a statement
that explicitly makes -1 an exceptional case. needless to say after
those first two examples i didn't bother trying to understand his last
two examples which were full of tell-tale pseudo-jargon like
"p-adically integral root vectors".
if you want to make a name for yourself as a prominent mathematical
crank then you'll at least have to try to outdo the guy that wrote
that book.
Sent via Deja.com http://www.deja.com/
Before you buy.
Mikal 606
"james dolan" <jdo...@math.ucr.edu> wrote in message
news:8o69k2$2b0$1...@nnrp1.deja.com...
Smile, James, you're on durian camera.
michael holmes
wrote:
>if you want to make a name for yourself as a prominent mathematical
>crank then you'll at least have to try to outdo the guy that wrote
>that book.
Are you that one guy who played Scotty on Star Trek?
Mike
John Conway
> sorry, you'll have to get in line behind all of the other cranks and >
> kooks before you who've made the same "discovery". just last week i >
> was reading a book where the author presents the idea that -1 is a >
> prime as his own brilliant idea. he gives a list of four example >
> statements that he claims are stated most concisely if -1 is >
> considered to be a prime. his first example is "every nonzero >
> rational number is uniquely a product of powers of prime numbers p"; >
> apparently he's too stupid to realize that -6 is both 2*3*(-1) and >
> 2*3*(-1)*(-1)*(-1).
I worded that statement more carefully than you've read it.
There is indeed a unique way that -6 is a product of powers of
distinct primes - namely those powers are -1, 2, and 3. There
are only two powers of -1, namely 1 and -1.
I remark that the fact that 24 is both 8*3 and 2*2*2*3
doesn't contradict the unique factorization theorem, since it
is understood that powers of the same prime are to be amalgamated.
his second example is "for distinct odd primes >
> f(p,q) and f(q,p) differ just when p = q = -1 (mod 4)", where f is the
> > "legendre symbol" function. this is just too brilliant for words: >
> he's trying to argue that -1 should be treated on the same footing as
> > the official genuine primes, and he does so by exhibiting a
> statement > that explicitly makes -1 an exceptional case.
I don't understand what you're saying here. My point was that
the case p = -1 does not behave exceptionally in the quadratic
reciprocity theorem, so does not need to be excepted. What statement
was being referred to in your last line?
> needless to
> say after > those first two examples i didn't bother trying to
> understand his last > two examples which were full of tell-tale
> pseudo-jargon like > "p-adically integral root vectors". > > if you
> want to make a name for yourself as a prominent mathematical > crank
> then you'll at least have to try to outdo the guy that wrote > that
> book. > >
A pity. If you'd continued to read the book, you might have
learned something. The citation for the Steele Prize for mathematical
exposition that I was awarded earlier this year said many complimentary
things about it, including that "ideas spill from every page". Your
folly has unfortunately prevented you from noticing them.
John Conway
John Conway
> - I worded that statement more carefully than you've read it.
and wrote:
> oh you did, did you????? that's funny, my copy doesn't say anything
> about the primes being distinct!!!!!! maybe you've been traveling
> around visiting all the libraries in the world and penciling in the
> word "distinct" and you just missed mine?????!!!!!!
No, not at all. I added that "distinct" for clarity, but went on to
remark later that it's usually understood.
It has, however, nothing to do with what I thought you were complaining
about, namely whether it's sensible to count -1 as a prime. If you
don't regard that "distinct" as understood, then the statement's false
whether we call -1 a prime or not, as is shown by my example 8*3
=2*2*2*3.
> - There is indeed a unique way that -6 is a product of powers of
> -distinct primes - namely those powers are -1, 2, and 3. There
> -are only two powers of -1, namely 1 and -1.
> -
> - I remark that the fact that 24 is both 8*3 and 2*2*2*3
> -doesn't contradict the unique factorization theorem, since it
> -is understood that powers of the same prime are to be amalgamated.
>
> the unique factorization theorem talks about factorizations into
> primes, not into prime powers, and it explicitly mentions the ordering
> ambiguity, leaving no unstated details that must be
> "understood"!!!!!!!
Let me freely admit that I should have added that word "distinct",
and perhaps said something about order, so that we can get back to
the real point at issue.
> - his second example is "for distinct odd primes
> -> f(p,q) and f(q,p) differ just when p = q = -1 (mod 4)", where f is
> the
> -> "legendre symbol" function. this is just too brilliant for words:
> -> he's trying to argue that -1 should be treated on the same footing as
> -> the official genuine primes, and he does so by exhibiting a
> -> statement that explicitly makes -1 an exceptional case.
> -
> - I don't understand what you're saying here. My point was that
> -the case p = -1 does not behave exceptionally in the quadratic
> -reciprocity theorem, so does not need to be excepted. What statement
> -was being referred to in your last line?
>
> ok i'll explain it once more, real slowly!!!!!!!! you see where it
> says "differ just when p = q = -1"?????? that sure doesn't sound
> like p = -1 behaving unexceptionally to me!!!!! you're explicitly
> saying that p = -1 is a special case!!!!!!!!
The exceptional case is when p and q are both congruent to -1
(mod 4). Are you saying that somehow this wasn't what was written
in the book?
Before I read on, let me say that I'm very puzzled indeed as to why my
calling -1 a prime so enrages you. I did so for a very real reason, not
just for fun. Namely, the theory of quadratic forms largely consists of
statements - let me call the typical one "Statement(p)" - that are
usually only asserted for the positive primes p = 2,3,5,... . Now in
my experience, in almost every case when Statement(p) is true for all
positive primes, it's automatically both meaningful and true for p = -1
too, and moreover, that additional assertion is just as useful.
Of course, it's also the case that -1 does behave differently to the
positive primes, but the way this happens is usually that MORE is true
when p = -1, not LESS. If LESS were true, then it would indeed be silly
to count -1 as a prime, but since it's almost always MORE, it's silly not
too.
Let me discuss the simplest such statements in this light.
First, the form of the statement of unique factorization that I
prefer is:
The multiplicative group of the non-zero rationals is the
direct product of the cyclic subgroups generated by "my" primes,
namely -1,2,3,5,... .
The additional thing that's true here is that the subgroup generated
by -1 has order 2.
Of course you can still state this with "your" primes, but then have
to say "generated by -1 and the primes", which makes it just a little
bit longer.
Second, my preferred statement of quadratic reciprocity is:
If p and q are distinct odd primes (in my sense), then
(p|q) and (q|p) differ precisely when p and q are both congruent
to -1 modulo 4.
If you want to make the same assertion without counting -1 as a prime,
then you have to add an additional clause:
" - moreover, (-1|p) is -1 precisely when p is congruent to -1
modulo 4 ".
Third, perhaps the best of my discoveries in that book is the fact
that for every odd "prime" p (in my sense), the number
p^a + [A|p^a] + p^b + [B|p^b] + ... (mod 8)
is an invariant of the quadratic form diag(p^a.A, p^b.B, ... )
(with the understanding that p^a is the p-part of p^a.A, etc.),
there being an analogous but slightly different statement for p = 2.
Once again, if you don't count -1 as a prime, the statement
becomes longer.
> -> needless to
> -> say after those first two examples i didn't bother trying to
> -> understand his last two examples which were full of tell-tale
> -> pseudo-jargon like "p-adically integral root vectors".
The fact that you misunderstood the second one suggests that you
stopped bothering even before that (but I apologize if indeed it's the
case that the "(mod 4)" condition in the second one was accidentally
misprinted).
The phrase "p-adically integral root vectors" is, unfortunately,
standard terminology, which I didn't have a good enough reason to change.
Judging by your absurd reaction to my proposal to count -1 as a prime,
I feel I was wise not to!
> oh, the steele prize, i'll bet that's a real important prize!!!!! let
> us know when you win a _real_ prize, like the nobel prize in
> mathematics!!!!!
Well, since you ask, this year I also accepted the $100,000 Nemmers
prize in mathematics. Unfortunately, there isn't a Nobel prize in
mathematics.
> i liked your complex analysis textbook, though.
Nice of you to say so, but I'm afraid it isn't mine, but
John B Conway's. I like it too, despite that fact!
John H Conway
PS. It really does puzzle me why you thought it so important to pour
scorn on the idea of counting -1 as a prime. All other readers have
somehow managed to control their rage at meeting this idea, and some
of them seem actually to have profited from the book! I'm sorry that
this minor matter has probably prevented you from doing so.
You seemed to find this idea stupid a priori, and so think me stupid
as well! I don't mind that at all; from my lofty position here such
things don't matter a damn. My point is that this a priori silly idea
is actually so sensible a posteriori that inside quadratic form theory
it's well worth while changing the accepted convention. But of course you
don't have to - by all means use the longer statements if you prefer them
- all this means is that I'll think you stupid too, and so we'll be quits!
Regards, JHC
John Conway
> I would like to state that I disagree with your tone in your reply to
> Professor Conway. If you have a point to make, then it is a mark of
> maturity and competence to be able to convey that point to anybody
> from the 5 year old child to a very well respective mathematician; and
> this is true even if you are diametrically opposite in viewpoints.
>
> Posting public statements like:>
>
> > ok i'll explain it once more, real slowly!!!!!!!! you see where it
> > says "differ just when p = q = -1"?????? that sure doesn't sound
> > like p = -1 behaving unexceptionally to me!!!!! you're explicitly
> > saying that p = -1 is a special case!!!!!!!!
>
> or
>
> > oh, the steele prize, i'll bet that's a real important prize!!!!!
> let
> > us know when you win a _real_ prize, like the nobel prize in
> > mathematics!!!!!
>
> do nothing to further you claim that you understand mathematics with
> greater insight than Professor Conway.
Thanks for this, Nim. However, in fact I seldom get offended by
such messages, since (as you suggest) they more clearly display the
failings of their senders than their targets. I did reply to this
one because I'm told that Mr Dolan is in fact a better mathematician
than his letters suggest. The distaste caused by his rudeness here
merely delayed my reply a day.
Regards,
John Conway
james dolan
- It has, however, nothing to do with what I thought you were
-complaining about, namely whether it's sensible to count -1 as a
-prime. If you don't regard that "distinct" as understood, then the
-statement's false whether we call -1 a prime or not, as is shown by my
-example 8*3 =2*2*2*3.
but if you don't count -1 as a prime then you get to state a cleaner
and stronger version of the unique factorization theorem
("factorization of positive integers into primes is unique up to
order") in which distinctness need not be mentioned at all.
but obviously your proposal to count -1 as a prime must be motivated
mainly by considerations other than how easy it is to state a unique
factorization theorem, and i'm in no position to have a serious
opinion about such other considerations yet, so i'm pretty much
completely neutral about your proposal. i do enjoy goofy ideas in
general, though, and when i saw beable van polasm trying to steal the
credit for your idea on sci.math i naturally rushed to defend your
priority in the matter.
so far i've only read the first lecture in the book and although it
was fun i didn't notice any evidence relevant to the "is -1 a prime?"
question. i'm not sure if the rest of the book is supposed to contain
any such evidence either, apart from the brief discussion in the
preface which wasn't detailed enough for me to completely understand
yet.
anyway, if there are any more editions i do recommend putting the word
"distinct" in in the appropriate place in the first italicized
sentence on page viii. also maybe the second italicized sentence
should be changed to "for distinct odd primes p and q, f(p,q) and
f(q,p) differ just when...". maybe it's true that your intended
audience is likely to be able to decipher what you meant in these
cases, but as long as you're fighting an uphill battle to convince
them of the superiority of your terminology you might as well treat
them as gently as possible.
- Third, perhaps the best of my discoveries in that book is the fact
-that for every odd "prime" p (in my sense), the number
-
- p^a + [A|p^a] + p^b + [B|p^b] + ... (mod 8)
-
-is an invariant of the quadratic form diag(p^a.A, p^b.B, ... )
-(with the understanding that p^a is the p-part of p^a.A, etc.),
-there being an analogous but slightly different statement for p = 2.
ah, thanks for spelling that out here. i'll have to think about it to
see how it affects my feelings about calling -1 a prime (not to
mention my feelings about calling 2 a prime).
- The fact that you misunderstood the second one suggests that you
-stopped bothering even before that (but I apologize if indeed it's the
-case that the "(mod 4)" condition in the second one was accidentally
-misprinted).
there was no misprint of that particular sort. (there may have been a
factual inaccuracy or two in my previous two posts; i don't actually
have much of an opinion about conway's complex analysis textbook for
example.)
-> oh, the steele prize, i'll bet that's a real important prize!!!!!
-> let us know when you win a _real_ prize, like the nobel prize in
-> mathematics!!!!!
-
- Well, since you ask, this year I also accepted the $100,000 Nemmers
-prize in mathematics. Unfortunately, there isn't a Nobel prize in
-mathematics.
could i borrow some money?
james dolan
-- Third, perhaps the best of my discoveries in that book is the fact
--that for every odd "prime" p (in my sense), the number
--
-- p^a + [A|p^a] + p^b + [B|p^b] + ... (mod 8)
--
--is an invariant of the quadratic form diag(p^a.A, p^b.B, ... )
--(with the understanding that p^a is the p-part of p^a.A, etc.),
--there being an analogous but slightly different statement for p = 2.
-
-ah, thanks for spelling that out here. i'll have to think about it to
-see how it affects my feelings about calling -1 a prime (not to
-mention my feelings about calling 2 a prime).
according to this week's "this week in the mathematics arxiv" posted
to sci.math.research by greg kuperberg, "there is a joke among
algebraic topologists that the only primes worth studying are 2 and
the infinite (sic) prime".
John Conway
> anyway, if there are any more editions i do recommend putting the word
> "distinct" in in the appropriate place in the first italicized
> sentence on page viii.
Well, I'll bear it in mind. But I'm not too repentant or too
worried, because after all the rigid style that would be appropriate
in the formal statement of a theorem definitely isn't for a mere
mention of it in the preface.
> ... but as long as you're fighting an uphill battle to convince
> them of the superiority of your terminology you might as well treat
> them as gently as possible.
I'm not fighting any uphill battle at all! In fact yours is
the only negative comment I've heard. In general, folks who've
read the book are quite happy with the idea, since they see its
very real advantages. I'm sure there must be a few who disagree,
but if so they obviously don't feel that it's a big enough deal
to be worth fighting over.
I don't either, for that matter. The advantages are clear
enough to ensure that I'll continue to use this convention when
working in this area, but I know that some more conservative
mathematicians won't. That's fine by me - I'll get the advantages,
and they won't!
> could i borrow some money?
I'm afraid not - all but $6000 of that prize went to pay
off my debts to the IRS, and that $6000 vanished into the
extra living expenses cause by my stay at Northwestern U
(who administer the Nemmers prizes). At the moment, I
have less than $300 in the bank!
John Conway
Li Dek
but does the above (fine I;m sure) book
suggests that from this day on we should replace the
"strong" unique factorization theorem: "factorization of positive
integers into primes is unique up to order" -[J.D]
with the "weaker" form:
" The multiplicative group of the non-zero rationals is the
direct product of the cyclic subgroups generated by "my" primes,
namely -1,2,3,5,... .
The additional thing that's true here is that the subgroup generated
by -1 has order 2. " -[J.C]
thanks in advance,
-Li