mm-1569
===
Subject: Re: discrete math
>I am thinking of taking discrete math class this quarter. Was just 
wondering
>how involved/difficult the class would be. I am already enrolled in a calc
>II class.. Would taking both be a bit too much?
Please don't post the same question multiple times.
The only possible answer is, it depends. Ask your college or the 
instructor how much homework and study time you should expect to 
spend for each class, and see if you have that time available.
-- 
Stan Brown, Oak Road Systems, Cortland County, New York, USA
                                        http://OakRoadSystems.com
In its default setup, Windows XP on the Internet amounts to a car
parked in a bad part of town, with the doors unlocked, the key in
the ignition and a Post-It note on the dashboard saying, 'Please
===
Subject: discrete math question
I am thinking of taking discrete math class this quarter. Was just 
wondering
how involved/difficult the class would be. I am already enrolled in a calc
II class.. Would taking both be a bit too much?
===
Subject: The point of diminishing returns.
To all Math Enthusiasts:
Time is very valuable to me. Accordingly, I would like to know if there is 
a
formula that I can use to calculate the total hours I can work before I
reach the point of diminishing returns. I would like this formula to
consider the following: Federal income tax, state tax, 401k as percentage 
of
gross income, and other deductions as a percentage of income. If anyone out
there can recommend a formula that I can use to calculate my personal point
of diminishing returns, it would be appreciated.
Ned
===
Subject: Re: The point of diminishing returns.
>Time is very valuable to me. Accordingly, I would like to know if there is 

a
>formula that I can use to calculate the total hours I can work before I
>reach the point of diminishing returns. I would like this formula to
>consider the following: Federal income tax,
What do you mean by point of diminishing returns? If you mean the 
point at which your marginal tax rate exceeds 100%, that happens 
only in very special circumstances (Social Security recipients with 
high-paying jobs and lots of interest income, IIRC). But ordinary 
working stiffs will not be in that position even if they work 168 
hours a week.
-- 
Stan Brown, Oak Road Systems, Cortland County, New York, USA
                                        http://OakRoadSystems.com
In its default setup, Windows XP on the Internet amounts to a car
parked in a bad part of town, with the doors unlocked, the key in
the ignition and a Post-It note on the dashboard saying, 'Please
Snore.
snip
plonk
RJ p
===
Subject: Re: Basis is a set of Eigenvector?
        boundary=----=_NextPart_000_001A_01C36F16.225AB740
---------------------------------------------------------------------
But, I have still some question of your reply.
that is following:
but, In my algebra book, the following is presented,
When T:V->V is a linear transformation, where {e 1,e 2,...,e n} is a basis 
of V,
if     T(e 1)=a 11*e 1 + a 12*e 2 + ... + a 1n*e n          then
       T(e 1)=a 21*e 1 + a 22*e 2 + ... + a 2n*e n
                                  ...
       T(e n)=a n1*e 1 + a n2*e 2 + ... + a nn*e n  
[T] = (row1,row2, ... ,row n)=(a 11 a 21 ... a n1 , a 12 a 22 ... a n2 , ... 

, a 1n a 2n ... a nn)
So, I thought A= (row1, row2, row3) = (0 0 0, 0 0 -1, 0 0 0) is true,  not 
(0 0 0, 0 0 0, 0 -1 0).
If my thinking is wrong, please explain why.
*sorry for using html*
  
===
Subject: Re: Teaching flawed information
> Are math professors willing to teach their students information they
> know is wrong?
<major snip>
I glanced at these threads of yours a few times.  I was immediately
suspicious of your claims but I never managed to summon the energy to
check the claims myself (supposing I first managed to make sense of
them).
But your recent thread on 1/3 being in the ring Z[1/2] is so
ridiculous that I did not need to expend any energy checking or trust
the opinions of others.  It was obviously completely wrong.  As many
others have pointed out, a ring does need to be closed under infinite
sums.  However my immediate reaction, which was hinted at by others
but not as far as noticed explicitly mentioned, was that a ring is not
required to have a topology or metric and an infinite sum cannot in
general be given a meaning.
You seem to only interested in rings which are sub-rings of the
complex numbers.  So all of your rings have an obvious topology.  But
there are many other rings and many of them do not have any standard
or interesting topology.  For example the integers modulo N (for some
positive natural number N) cannot be given an interesting topology. 
Hence infinite sums are not possible or interesting in these rings.  I
am quite sure that nonetheless they are rings.
I now know, just from my own readings, that there is no significant
chance that anything you post is likely to be valid or interesting
mathematics.
You have not damaged my faith in mathematics or its teaching.  Heck,
even if you were right, how many maths courses even mention algebraic
integers, so how many courses and teachers would be invalidated.  I
already knew that bad maths teachers existed, some making far worse
errors.  But fortunately I have also met some very good ones.
The FBI nonsense is absolutely incredible.  Fortunately I am outside
their jurisdiction so I can continue to believe that the algebraic
integers are a ring and 1/3 is not in the ring Z[1/2] without fear of
arrest.
J
===
Subject: Re: Teaching flawed information
>Are math professors willing to teach their students information they
>know is wrong?
> <major snip>
> I glanced at these threads of yours a few times.  I was immediately
> suspicious of your claims but I never managed to summon the energy to
> check the claims myself (supposing I first managed to make sense of
> them).
Well given your low energy, I'll help you out by giving them to you.
I've found the prime counting function that gets primes as the result
of integrating a partial difference equation.  It has a partial
differential equation analog, which from my experiments numerically
integrates to give a curve that is closer to the prime distribution
than Li(x) but further than R(x), the Riemann function, so it may be
THE reason for something that has intrigued mathematicians since
Gauss.
Also I found a short proof of Fermat's Last Theorem which uses mostly
rather basic algebraic methods.  It is well critiqued with intense
opposition being over a few lines, which I've explained in detail over
a period of more than a year.  That's a lead-in to the next important
point.
You see, my third major result is that rather than keep arguing with
people about my FLT proof, I took the methods, which had faced so much
opposition, and used them to show an error in core mathematics,
which is a problem with the ring of algebraic integers.  That allows
me both to refute the objections to my FLT proof, and further show
just how extraordinarily powerful my methods are.  Further it put
mathematicians in the position of either admitting the truth or
engaging in fraud, by refusing to acknowledge the error so that they
can continue to teach students false information.
 
> But your recent thread on 1/3 being in the ring Z[1/2] is so
> ridiculous that I did not need to expend any energy checking or trust
> the opinions of others.  It was obviously completely wrong.  As many
> others have pointed out, a ring does need to be closed under infinite
> sums.  However my immediate reaction, which was hinted at by others
> but not as far as noticed explicitly mentioned, was that a ring is not
> required to have a topology or metric and an infinite sum cannot in
> general be given a meaning.
Well that sounds like you have a sense of decidability and your
intuition is that infinite sums aren't always decidable in a ring.
That is true.  For instance, in integers 1+1+1+... is not decidable,
as infinity is not a number.
However, 1/3 = 1/2 + 1/2^2 +1/2^3 +... is decidable, and in fact
there's no way to specifically exclude it by the definition of a ring.
What I've noticed is that posters cheat, as they *want* to exclude the
result, so they start talking but never actually make any sense, as
you  begin to realize that really they just want to exclude the
result.
That is, mathematically objections I've seen to including 1/3 in
Z[1/2] are empty, but make sense if you realize that the posters just
WANT to exclude 1/3 so badly that really they've just decided it's
excluded and don't give a damn what the math actually says.
 
> You seem to only interested in rings which are sub-rings of the
> complex numbers.  So all of your rings have an obvious topology.  But
> there are many other rings and many of them do not have any standard
> or interesting topology.  For example the integers modulo N (for some
> positive natural number N) cannot be given an interesting topology. 
> Hence infinite sums are not possible or interesting in these rings.  I
> am quite sure that nonetheless they are rings.
Well you just said that infinite sums are not possible in finite
rings.
Duh.
Again, what is clear from your post is that you do NOT want 1/3
included in Z[1/2] and the math be damned.
 
> I now know, just from my own readings, that there is no significant
> chance that anything you post is likely to be valid or interesting
> mathematics.
> You have not damaged my faith in mathematics or its teaching.  Heck,
> even if you were right, how many maths courses even mention algebraic
> integers, so how many courses and teachers would be invalidated.  I
> already knew that bad maths teachers existed, some making far worse
> errors.  But fortunately I have also met some very good ones.
I was a physics undergrad, and for me and my fellow physics
undergrads, it was extremely important that what we were told was the
truth to the limits of the professors ability to give us the truth.
But you see in physics, it's understood that learning is a continual
process which involves constant checking at all levels.
Mathematicians have instead opted for a coral process, where what is
built upon is assumed to be perfect, when math students should be
taught to challenge.
> The FBI nonsense is absolutely incredible.  Fortunately I am outside
> their jurisdiction so I can continue to believe that the algebraic
> integers are a ring and 1/3 is not in the ring Z[1/2] without fear of
> arrest.
Mathematicians are important members of the security apparatus of the
United States, and it is VERY important if there is a question of the
integrity of the group.
After all, mathematicians may start willing to lie about important
math results, and having proven to be corrupt, later move to more
directly acting against the interests of the United States of America.
It is vital that the field be held to a very high standard, which
includes a very high *moral* standard.
James Harris
===
Subject: Re: Teaching flawed information
> It is vital that the field be held to a very high standard, which
> includes a very high *moral* standard.
How could a thief, liar and charlatan like you know *anything* about
moral standards?
-- 
Wayne Brown                | When your tail's in a crack, you improvise
fwbrown@bellsouth.net      |  if you're good enough.  Otherwise you give
                           |  your pelt to the trapper.
e^(i*pi) = -1  -- Euler  |           -- John Myers Myers, 
Silverlock
===
Subject: Re: Teaching flawed information
            Visiting Assistant Professor at the University of Montana.
>> It is vital that the field be held to a very high standard, which
>> includes a very high *moral* standard.
>How could a thief, liar and charlatan like you know *anything* about
>moral standards?
How else who he avoid them so completely and so successfully for such
a long period of time?
 [Gabriele Rossetti] has left a vast body of writings... in which
  he has attempted to prove the truth of his unorthodox interpre-
  tation of medieval literature. They present a formidable
  record of unsystematic research in which we see an enthusiast
  plunging farther and farther and farther from the logic of facts
  and good sense until truth is lost in the dreadful nightmare
  of an idee fixe. There is no real evolution of the Theory
  although it grows and expands until it embraces ever wider
  horizons. The numerous inaccuracies of deduction, mis-statements
  of historical fact, and self-contradictions...have caused critics
  to turn awy from them in disgust... [...] It is impossible to
  read far... without realizing that we have to deal with a work of
  faith and imagination rather than of reasoning. There is an
  appearance of reason, for the author is set on proving by logic
  the truth of what he already believes by intuition. The truth
  is plain to him and he cannot comprehend why others do not
  immediately accept it, but as they desire demonstration he has
  multiplied his proofs. It is the redundancy and confusion of a
  prophet expounding by a familiar method the truth revealed to his
  own simple soul in a flash of inspiration... In such work as 
  this... it is idle to look for the calm reasoning of a scholar;
  we do not find it, and there is little or no advantage in
  attacking the obvious inconsistencies and absurdities that abound.
     -- E.R. Vincent, _Gabriele Rossetti in England_, quoted in
        _The Shakespearan Ciphers Examined_, by William F.
        Friedman and Elizebeth S. Friedman
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Teaching flawed information
                ... stuff deleted ...
>>But your recent thread on 1/3 being in the ring Z[1/2] is so
>>ridiculous that I did not need to expend any energy checking or trust
>>the opinions of others.  It was obviously completely wrong.  As many
>>others have pointed out, a ring does need to be closed under infinite
>>sums.  However my immediate reaction, which was hinted at by others
>>but not as far as noticed explicitly mentioned, was that a ring is not
>>required to have a topology or metric and an infinite sum cannot in
>>general be given a meaning.
> Well that sounds like you have a sense of decidability and your
> intuition is that infinite sums aren't always decidable in a ring.
> That is true.  For instance, in integers 1+1+1+... is not decidable,
> as infinity is not a number.
> However, 1/3 = 1/2 + 1/2^2 +1/2^3 +... is decidable, and in fact
> there's no way to specifically exclude it by the definition of a ring.
> What I've noticed is that posters cheat, as they *want* to exclude the
> result, so they start talking but never actually make any sense, as
> you  begin to realize that really they just want to exclude the
> result.
The issue of who *wants* to exclude what is all your perverse
fabrication. Mathematics isn't about someone *wanting* anything.
In mathematics, one has definitions, and then proves properties.
> That is, mathematically objections I've seen to including 1/3 in
> Z[1/2] are empty, but make sense if you realize that the posters just
> WANT to exclude 1/3 so badly that really they've just decided it's
> excluded and don't give a damn what the math actually says.
Let's see who gives a damn about what the math actually says:
Please produce the following:
        1. The definition of the ring Z[x].
        2. The definition of the ring Z[1/2]. This definition
           must be consistent with the definition in 1 above.
        3. A proof from the the definition of Z[1/2] that this
           ring contains the number 1/3. You must use only the
           properties of the ring given in the definition from
           item 2, or those which have been proven directly from
           this definition. For instance, if you care to use
           the convergence (or as you're incorrectly terming it,
           decidability) of a sequence or series, then you must
           prove that the ring is closed under the passage to
           limits.
I have no doubt that you will fail to do this, and will explain it as
somehow wrong for mathematicians even to suggest that you perform these
little tasks.  I, for one, and I am certain that many others who are
participating in these threads, can do 1 and 2 above, and can also show
that 3 above is *incorrect*.
The fact that you cannot, and will not, is more than ample evidence of
your failure to produce mathematics.
>>You seem to only interested in rings which are sub-rings of the
>>complex numbers.  So all of your rings have an obvious topology.  But
>>there are many other rings and many of them do not have any standard
>>or interesting topology.  For example the integers modulo N (for some
>>positive natural number N) cannot be given an interesting topology. 
>>Hence infinite sums are not possible or interesting in these rings.  I
>>am quite sure that nonetheless they are rings.
> Well you just said that infinite sums are not possible in finite
> rings.
> Duh.
> Again, what is clear from your post is that you do NOT want 1/3
> included in Z[1/2] and the math be damned.
Again, that's not math. The term want is NOT a mathematical term,
is it? Just who is it who is inserting emotional language (viz, WANT)
into this discussion?
>>I now know, just from my own readings, that there is no significant
>>chance that anything you post is likely to be valid or interesting
>>mathematics.
>>You have not damaged my faith in mathematics or its teaching.  Heck,
>>even if you were right, how many maths courses even mention algebraic
>>integers, so how many courses and teachers would be invalidated.  I
>>already knew that bad maths teachers existed, some making far worse
>>errors.  But fortunately I have also met some very good ones.
> I was a physics undergrad, and for me and my fellow physics
> undergrads, it was extremely important that what we were told was the
> truth to the limits of the professors ability to give us the truth.
And there you have it. Your professors
        Told you the truth to the limits of
        their ability to give you the truth.
My professors
        Gave us the definitions, and PROVED
        what they claimed. If a student claimed
        not to follow a step in a proof, then
        that step was fully justified (either
        at that time, or noted at the time as
        a CLAIM, and then all CLAIMS were proven
        at the end of the main proof).
Can you see the difference? Your folks TOLD you what they claimed, my
folks PROVED what they claimed.
        TOLD
        PROVED.
Two different words. Two different meanings. But we all know about your
comprehension problems regarding your native language. It's not a pretty
picture, being so bereft of facility in one's native tongue, and maybe
you should get some help in that account.
> But you see in physics, it's understood that learning is a continual
> process which involves constant checking at all levels.
Constant checking at all levels.  Somehow, you imagine that proof is
less certain than constant checking at all levels. Ain't that special?
> Mathematicians have instead opted for a coral process, where what is
> built upon is assumed to be perfect, when math students should be
> taught to challenge.
You have no concept of what mathematicians go through to learn their
craft.  The language you use is suggestive of your continues claims
that we are told what is true, and assume it's perfect. Nothing
could be farther from the truth, and you have been told as much. Short
of having you actually *take* a math class (which you've avoided like
the worst poison), there is no way to show you the truth. Therefore,
you are more than willing to be twice the bigot that you accuse others
of being.
>>The FBI nonsense is absolutely incredible.  Fortunately I am outside
>>their jurisdiction so I can continue to believe that the algebraic
>>integers are a ring and 1/3 is not in the ring Z[1/2] without fear of
>>arrest.
> Mathematicians are important members of the security apparatus of the
> United States, and it is VERY important if there is a question of the
> integrity of the group.
I'm a member of NO apparatus. By the way, which apparatus are YOU a
member of? The humor apparatus? The bigotry apparatus? The punk
apparatus? The Hip-Hop apparatus? The drunken fool apparatus?
> After all, mathematicians may start willing to lie about important
> math results, and having proven to be corrupt, later move to more
> directly acting against the interests of the United States of America.
I'd look around to see if any of that spittle spashed onto your keyboard
or monitor screen at this point. What with all the foaming at the mouth,
you may want to take to wearing a spit shield, for hygienic purposes.
> It is vital that the field be held to a very high standard, which
> includes a very high *moral* standard.
And you ask why people are so energetic in opposing you?  It is
precisely the maintenance of standards. Not very high standards, but
absolute standards. Further, it is your contribution that opposes the
standards of proof that your critics maintain.
For you to suggest that *YOUR* mathematical standards are in any way
more stringent than those of your critics is ridiculous.  For you to
suggest that your critics are lying is shameless.  However, for you
to go to the FBI or CIA or NSA or the US Army, well, that's PRICELESS.
> James Harris
Dale.
===
Subject: Re: Teaching flawed information
so, what are you going to do, if he doesn't oblige --
what are *we* going to do, call hte FBI?
    that would probably be the safest recourse, although,
since he seems to be working for them, the CIA might
be a better initiating agency.  consider that
he's developed a great facility for elementary math hijinx and
(the greater portion of his 8-year programme) personal grandiloguence,
and how is the average Joe Agent supposed to not take him seriously?
    the thing about Wiles' efforts is not that they're so un-
elementary, but that they tie Fermat's Last theorem
into so many other parts of mathematics.  if
we don't get Harris and his Golden Key into the NSA, soon,
we could lose him to the Russians ... or the British ... or
the Austro-Hungarian Empire under Ahnold!
seriously, you may have hit his sorest point at Vanderbilt, although
*most* of the mainstream USA curriculum is weighted
by the dead hand of Newton, empiricism etc. ad vomitorium. (see
 
> Please produce the following:
>       1. The definition of the ring Z[x].
 
> I have no doubt that you will fail to do this, and will explain it as
> somehow wrong for mathematicians even to suggest that you perform these
> little tasks.  I, for one, and I am certain that many others who are
> participating in these threads, can do 1 and 2 above, and can also show
> that 3 above is *incorrect*.
 
>I was a physics undergrad, and for me and my fellow physics
>undergrads, it was extremely important that what we were told was the
>truth to the limits of the professors ability to give us the truth.
--A church-school McCrusade (Blair's ideals?):
Harry-the-Mad-Potter want's US to kill Iraqis?...
http://www.tarpley.net/bush25.htm (Thyroid Storm ch.)
  http://www.rwgrayprojects.com/synergetics/plates/plates.html
===
Subject: Re: Teaching flawed information
<deleted>
> 
>> 
>>   You claim that if you have a polynomial of the form
>> 
>> 
>>    P(x) = (v^3 + 1)*x^3 - 3*v*x*(u*f)^2 + (u*f)^3,
>> 
>> 
>> where v = -1 + m*f^2, and m, u, and f are integers,
>> with f prime and m coprime to f, then P(x)/f^2 
>> can be factored in the form
>> 
>> 
>>    P(x)/f^2 = (b1*x + u)*(b2*x + u)*(b3*x + u*f)    [1]
>> 
>> 
>> where b1, b2, and b3 are algebraic integers.
>
>No I don't.
>
>   I downloaded Advanced Polynomial Factorization 
> from your website last night.  You said, assuming
>      P(x) = (a1*x + uf)*(a2*x + uf)*(a3*x + uf),
> where a1, a2, and a3 are algebraic integers, then
> letting g1 = a1*x + uf, g2 and g3 defined similarly,
>   two of the g's should have a factor of f which
>    would force two of the a's to have a factor
>    that is f.
> Assume without loss of generality that a1 
> and a2 have a factor that is f.
Yup, two of the g's *should* have a factor of f.
 
> To equate this to what I said above, let
>    b1 = a1/f, and b2 = a2/f.
> Since you say essentially that a1 and a2
> have a factor that is f, this means that
> b1 and b2 are algebraic integers.
They should be algebraic integers.
>   The question here is: why are you denying the
> results in your own paper? 
Nope.
>The problem is that the ring of algebraic integers doesn't include
>them.
Which goes to why I use the word should as I do.
<deleted>
>The problem is that the b's aren't included in the ring of algebraic
>integers, which I've pointed out more than once.
>
>   See above.  When you say two of the a's have a 
> factor that is f, and you are working in the ring
> of algebraic integers, that can mean only one thing:
> b1 = a1/f and b2 = a2/f are algebraic integers.
>   Or are you now saying that your results in APF
> are incorrect ???
Nope.
>So how is such an error a moral test?
>
Here the poster Nora Baron has done something done before which is
to delete out context.
The test is how much society cares for its students, with the queston
of whether or not it will allow mathematicians to teach flawed
information.
I'm curious about that question.
>   Yes, perhaps it is.
>   Nora B.
It is a moral question which goes to the heart of academia.
 
>Mathematicians might believe they can sit back with my prime counting
>function or my proof of Fermat's Last Theorem, but ignoring an error
>requires allowing it to be taught to students.
>
>My assessment is that mathematicians lack the morals to make the right
>decision, but more interesting to me at this point is whether or not
>society will allow mathematicians to proceed with teaching flawed
>information.
>
>That is, will humanity protect its children in this area?
The answer is, yes.
James Harris
===
Subject: Re: Teaching flawed information
            Visiting Assistant Professor at the University of Montana.
><deleted>
>> 
> 
>   You claim that if you have a polynomial of the form
> 
> 
>    P(x) = (v^3 + 1)*x^3 - 3*v*x*(u*f)^2 + (u*f)^3,
> 
> 
> where v = -1 + m*f^2, and m, u, and f are integers,
> with f prime and m coprime to f, then P(x)/f^2 
> can be factored in the form
> 
> 
>    P(x)/f^2 = (b1*x + u)*(b2*x + u)*(b3*x + u*f)    [1]
> 
> 
> where b1, b2, and b3 are algebraic integers.
>>
>>No I don't.
>>
>>   I downloaded Advanced Polynomial Factorization 
>> from your website last night.  You said, assuming
>>      P(x) = (a1*x + uf)*(a2*x + uf)*(a3*x + uf),
>> where a1, a2, and a3 are algebraic integers, then
>> letting g1 = a1*x + uf, g2 and g3 defined similarly,
>>   two of the g's should have a factor of f which
>>    would force two of the a's to have a factor
>>    that is f.
>> Assume without loss of generality that a1 
>> and a2 have a factor that is f.
>Yup, two of the g's *should* have a factor of f.
In other words: all your complaints about the algebraic integers being
broken or flawed really mean the algebraic integers do not have
the property of magically making my argument correct.
 [Gabriele Rossetti] has left a vast body of writings... in which
  he has attempted to prove the truth of his unorthodox interpre-
  tation of medieval literature. They present a formidable
  record of unsystematic research in which we see an enthusiast
  plunging farther and farther and farther from the logic of facts
  and good sense until truth is lost in the dreadful nightmare
  of an idee fixe. There is no real evolution of the Theory
  although it grows and expands until it embraces ever wider
  horizons. The numerous inaccuracies of deduction, mis-statements
  of historical fact, and self-contradictions...have caused critics
  to turn awy from them in disgust... [...] It is impossible to
  read far... without realizing that we have to deal with a work of
  faith and imagination rather than of reasoning. There is an
  appearance of reason, for the author is set on proving by logic
  the truth of what he already believes by intuition. The truth
  is plain to him and he cannot comprehend why others do not
  immediately accept it, but as they desire demonstration he has
  multiplied his proofs. It is the redundancy and confusion of a
  prophet expounding by a familiar method the truth revealed to his
  own simple soul in a flash of inspiration... In such work as 
  this... it is idle to look for the calm reasoning of a scholar;
  we do not find it, and there is little or no advantage in
  attacking the obvious inconsistencies and absurdities that abound.
     -- E.R. Vincent, _Gabriele Rossetti in England_, quoted in
        _The Shakespearan Ciphers Examined_, by William F.
        Friedman and Elizebeth S. Friedman
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Teaching flawed information
> <deleted>
>>  
>>
>>  You claim that if you have a polynomial of the form
>>
>>
>>   P(x) = (v^3 + 1)*x^3 - 3*v*x*(u*f)^2 + (u*f)^3,
>>
>>
>>where v = -1 + m*f^2, and m, u, and f are integers,
>>with f prime and m coprime to f, then P(x)/f^2 
>>can be factored in the form
>>
>>
>>   P(x)/f^2 = (b1*x + u)*(b2*x + u)*(b3*x + u*f)    [1]
>>
>>
>>where b1, b2, and b3 are algebraic integers.
>> 
>> No I don't.
>> 
>
>
>  I downloaded Advanced Polynomial Factorization 
>from your website last night.  You said, assuming
>
>
>     P(x) = (a1*x + uf)*(a2*x + uf)*(a3*x + uf),
>
>where a1, a2, and a3 are algebraic integers, then
>letting g1 = a1*x + uf, g2 and g3 defined similarly,
>
>  two of the g's should have a factor of f which
>   would force two of the a's to have a factor
>   that is f.
>
>Assume without loss of generality that a1 
>and a2 have a factor that is f.
> Yup, two of the g's *should* have a factor of f.
  Yes, that is the wording in APF.  So what you
must now be saying is that they *should* have a
factor that is f, but they do not.  Right?
  See also below.
  
>To equate this to what I said above, let
>
>   b1 = a1/f, and b2 = a2/f.
>
>Since you say essentially that a1 and a2
>have a factor that is f, this means that
>b1 and b2 are algebraic integers.
> They should be algebraic integers.
  Aha.  Should again.
>  The question here is: why are you denying the
>results in your own paper? 
> Nope.
>> The problem is that the ring of algebraic integers doesn't include
>> them.
> Which goes to why I use the word should as I do.
  And should yet again.
> <deleted>
>> The problem is that the b's aren't included in the ring of algebraic
>> integers, which I've pointed out more than once.
>> 
>
>  See above.  When you say two of the a's have a 
>factor that is f, and you are working in the ring
>of algebraic integers, that can mean only one thing:
>b1 = a1/f and b2 = a2/f are algebraic integers.
>
>  Or are you now saying that your results in APF
>are incorrect ???
> Nope.
  Right.  Not incorrect because you said should 
rather than, e.g., must.
  I think what we have here is the following.  You
are saying should because you now know that in
fact b1 and b2 cannot be algebraic integers.  In 
other words, you have somehow arrived at the same
point as the rest of us.
  But we do not agree on what one must conclude 
from this.  I conclude that your original claim,
in the original version of APF, was incorrect and
that your proof was in error.  You conclude now,
apparently, that you have arrived at a basic
contradiction: elementary algebraic number theory
shows that b1 and b2 cannot be algebraic integers,
but you think they *must* be, therefore algebraic
number theory is wrong somehow.  You have phrased 
this as, the ring of algebraic integers is
incomplete.  But that is just dressing it up.
If you were right, you would have discovered a
fundamental inconsistency in mathematics.
  Evidently you are not willing to bite that 
particular bullet.
  It still gets back to the word should.  In
order to justify should, you have to return to
your argument about m = 0 (the degenerate case)
and f = 3 (the case where in general the associated
polynomials are not irreducible).  We have 
discussed flaws in your reasoning for both of these
cases in detail and I do not agree with your
conclusions.  *Even if I did*, I do not see how
showing a pattern for m = 0 and for f = 3
proves anything about f = prime > 3 and m coprime
to f.  For example, how would a statement for
m = 0 and f = 3 tell me anything about m = 1
and f = 5?  I think even you must admit that that part
of your argument is missing.  All you have at 
best is circumstantial evidence for two 
peripheral cases.  That evidence, even if one
believed it, would not constitute a proof.
  But this is your basis for saying should.  
>
>> So how is such an error a moral test?
>> 
> Here the poster Nora Baron has done something done before which is
> to delete out context.
  I left in the *math* context.  I deleted out the
pontificating and social commentary.  
> The test is how much society cares for its students, with the queston
> of whether or not it will allow mathematicians to teach flawed
> information.
> I'm curious about that question.
  I disagree with your conclusion.  I have good, logical
mathematical grounds for that disagreement.  If you want
to discuss the math and the logic I am happy to do so.
I am not interested in the moralizing.  After all, if you
have not in fact proved what you claim mathematically, then
all your talk of mathematicians teaching flawed information
is just hot air.  You must first rigorously prove that we
are wrong before you can get on a soapbox and claim we 
teach the wrong math.  
  You have no such rigorous proof.
  
>  Yes, perhaps it is.
>
>  Nora B.
> It is a moral question which goes to the heart of academia.
  Not until you have a proof that you are right.
And at the moment, as at all previous moments, 
you are empty-handed.
  Nora B.
>> Mathematicians might believe they can sit back with my prime counting
>> function or my proof of Fermat's Last Theorem, but ignoring an error
>> requires allowing it to be taught to students.
>> 
>> My assessment is that mathematicians lack the morals to make the 
right
>> decision, but more interesting to me at this point is whether or not
>> society will allow mathematicians to proceed with teaching flawed
>> information.
>> 
>> That is, will humanity protect its children in this area?
> The answer is, yes.
> James Harris