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| Author |
Message |
jp2
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The Never Ending Proof
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May 4 19:28 UTC 2002 |
This item has been erased.
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| 81 responses total. |
aruba
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response 1 of 81:
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May 4 22:30 UTC 2002 |
Then there is a y such that xy = 1.
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jp2
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response 2 of 81:
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May 4 22:32 UTC 2002 |
This response has been erased.
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rcurl
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response 3 of 81:
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May 5 01:00 UTC 2002 |
What is y !? I know its not factorial.
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brighn
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response 4 of 81:
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May 5 04:43 UTC 2002 |
y = 1/x ... isn't that enough?
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i
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response 5 of 81:
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May 12 04:33 UTC 2002 |
Re: #3
"y != 0" is "y is not equal to zero". The usual "not equal" symbol is
much harder to do consistently across computer platforms.
x and y are either both irrational or both rational.
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utv
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response 6 of 81:
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May 12 15:13 UTC 2002 |
if x and y are mail parcels, they are both irradiated.
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jp2
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response 7 of 81:
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May 12 15:52 UTC 2002 |
This response has been erased.
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albaugh
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response 8 of 81:
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May 13 21:31 UTC 2002 |
xyy < normal
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shashee
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response 9 of 81:
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May 16 14:58 UTC 2002 |
Either xyx < 1 or yxy < 1
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jp2
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response 10 of 81:
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May 16 15:26 UTC 2002 |
This response has been erased.
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aruba
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response 11 of 81:
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May 16 15:38 UTC 2002 |
Kind of obvious, I would have thought.
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jp2
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response 12 of 81:
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May 16 16:24 UTC 2002 |
This response has been erased.
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remmers
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response 13 of 81:
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May 16 18:03 UTC 2002 |
Okay, then let's try a different game.
The person who's "it" posts a mathematical fact. The first person
to supply a proof acceptable to the poster is now "it" and gets to
enter the next fact.
In the interest of keeping the game moving and also keeping it
interesting, the facts should be neither too trivial nor too
difficult to prove. In particular, avoid unsolved conjectures.
I'll start: Prove that there are infinitely many prime numbers.
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jp2
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response 14 of 81:
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May 16 18:06 UTC 2002 |
This response has been erased.
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remmers
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response 15 of 81:
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May 16 18:08 UTC 2002 |
I'll accept that. Go!
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jp2
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response 16 of 81:
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May 16 18:18 UTC 2002 |
This response has been erased.
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flem
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response 17 of 81:
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May 16 20:01 UTC 2002 |
Hmm. I believe integers modulo, say, 5 will do. ISTR that integers modulo
any prime will do, but don't recall how to go about proving it. w.r.t.
integers mod 5, it suffices to observe that 3 = 4 * 2 mod 5.
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jp2
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response 18 of 81:
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May 16 20:21 UTC 2002 |
This response has been erased.
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flem
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response 19 of 81:
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May 17 14:58 UTC 2002 |
Yeah, but I couldn't remember the exact definition of prime within a ring,
so I went for an example that I thought would work with a weaker definition.
My turn, hm? Well, what the heck, let's generalize:
Definitions:
An element of a ring is a unit if it has a multiplicative inverse.
An element p of a ring is prime if it is not a unit and if it cannot
be decomposed into factors p = ab, where neither a nor b is a unit.
Claim: Let p be a positive prime integer. Then the ring Z/pZ (integers
modulo p) contains no prime elements.
(I'm not entirely happy with these definitions and this statement of
the claim, but I don't have my books with me and that's the best I can
do in ten minutes on the internet. Feel free to improve.)
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aruba
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response 20 of 81:
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May 17 15:53 UTC 2002 |
Hungerford says:
An element c of a ring R is *irreducible* if
(i) c is a nonzero nonunit
(ii) c = ab ==> a or b is a unit.
An element p of a ring R is *prime* if
(i) p is a nonzero nonunit
(ii) p | ab ==> p|a or p|b.
In Z/pZ, primes and irreducibles are the same thing. In a general domain
(a ring where ab = 0 ==> a=0 or b=0), primes are irreducible, but
irreducibles are not necessarily prime.
(As a side note, I first saw Red Dwarf about the time I learned that, and
for a long time, I heard the beginning of the theme song as,
"It's cold outside, primes are irreducible"
Dunno why.)
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bhelliom
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response 21 of 81:
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May 17 17:38 UTC 2002 |
My brain hurts. <sulks> Truly, as a non-math person, I'm impressed.
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flem
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response 22 of 81:
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May 17 17:52 UTC 2002 |
Hungerford is a good book. I used a library copy for a couple of months when
I was a research assistant. I've wished a couple of times since then that
I owned a copy.
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utv
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response 23 of 81:
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May 17 23:52 UTC 2002 |
didn't tolkien write THEbook on Ring Theory.
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gelinas
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response 24 of 81:
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May 18 02:54 UTC 2002 |
I found 19 and 20 completely incomprehensible.
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