mm-1016
===
Subject: Fermatproof of FLT
Stay tune for presentation of Fermats proof of FLT
No beating the bush around!
As Euler miss to find a elementary proof of 
Pells Equation so
it
happened that another great mathematicians miss to find the
Fermats proof of FLT.
By tomorow morning(may be tonight)will be posted.
george ghiata
===
Subject: Re: Fermatproof of FLT
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBL3uDj24024;
>Stay tune for presentation of Fermats proof of FLT
>No beating the bush around!
>As Euler miss to find a elementary proof of 
Pells Equation
so it
>happened that another great mathematicians miss to find the
>Fermats proof of FLT.
>By tomorow morning(may be tonight)will be posted.
>george ghiata
I cant wait until tomorrow morning, please post tonight.
Joseph A.
===
Subject: Re: Fermatproof of FLT
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLFfMi18716;
Hi Joseph,
I am sorry,
I want to let all of you that I was very late back home and I
just
wake -up .I do not want to let you down but when I recover I
will
sit down and write the message and post it right the way.
That will be very soon.
P.S.So there is left a chance for somebodyelse too.
george ghiata
>>Stay tune for presentation of Fermats proof of FLT
>>No beating the bush around!
>>As Euler miss to find a elementary proof of 
Pells Equation
so it
>>happened that another great mathematicians miss to find the
>>Fermats proof of FLT.
>>By tomorow morning(may be tonight)will be posted.
>>george ghiata
>I cant wait until tomorrow morning, please post tonight.
>Joseph A.
===
Subject: Re: And then there was George ...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBKLliD25147;
Well.from your point of you the odds are against me.
I recognise that.
george
>>Stay tune for presentation of Fermats proof of FLT
>>No beating the bush around!
>>As Euler miss to find a elementary proof of 
Pells Equation
so it
>>happened that another great mathematicians miss to find the
>>Fermats proof of FLT.
>>By tomorow morning(may be tonight)will be posted.
>>george ghiata
>George,
>I was wondering if you had any notions regarding 0.999...?
>- MO
===
Subject: Re: Why do two negatives equal a positive?
> using the field axioms is perfect except for a major ßaw
which
> (atleast in my eyes) has rendered it useless. we are trying
to prove
> here that when a negetive and a positive integer is
multiplied the
> result is positive.
No, we are trying to show that when two negative integers are
multiplied
the
result is positive. Multiplying a positive and a negative
will give you a
negative...
> why? but in your solution, you have used the same
> property in the third line without proving it first.
Here are the second and third lines, if I understand you
correctly:
So where exactly has Matt used the same property without
proof?
(Answer: he hasnt!)
And BTW many people consider top-posting as you have done to
be rude (just
to warn you...)
Mike.
>By using the field axioms
>(<a
href=http://mathworld.wolfram.com/FieldAxioms.html>http://
mathworld.wolf
ra
m.com/FieldAxioms.html</a>)
>first you can show (-1)(-1) = 1:
>(-1)(-1) = (-1)(-1) + 0
> = (-1)(-1) + (-1) + 1
> = (-1)[(-1) + 1] + 1
> = (-1) * 0 + 1
> = 0 + 1 = 1
>and that (-1)a = (-a):
>(-1)a = (-1)a + 0
> = (-1)a + a + (-a)
> = a[(-1) + 1] + (-a)
> = a * 0 + (-a)
> = 0 + (-a) = (-a)
>So (-a)(-b) = (-1)a * (-1)b
> = (-1)(-1)ab
> = 1 * ab = ab.
===
Subject: Riddle to solve....
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBL2R6O17138;
I am in the sea BUT not in the water
I start in March and end in December
You find me neither in peace nor war
Many have me but few dont
I am in the shadows but not in a dark place
===
Subject: Re: Riddle to solve....
> I am in the sea BUT not in the water
> I start in March and end in December
> You find me neither in peace nor war
> Many have me but few dont
> I am in the shadows but not in a dark place
Please post riddles in the rec.puzzles news group rather than
a
mathematical group.
Ken Pledger.
===
Subject: Please help
Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at -2
and a local min value of 0 at 1.
--
submissions: post to k12.ed.math or e-mail to
k12math@k12groups.org
private e-mail to the k12.ed.math moderator:
kem-moderator@k12groups.org
newsgroup website: http://www.thinkspot.net/k12math/
newsgroup charter:
http://www.thinkspot.net/k12math/charter.html
===
Subject: Re: Please help
> Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
> and a local min value of 0 at 1.
Cant be done as stated. Coefficient of x^3 
cant be 9.
If you mean f(x) = a x^3 + b x^2 + c x + d, then it becomes
possible.
===
Subject: Re: Please help
> Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3
> at -2
> and a local min value of 0 at 1.
I think you meant f(x) = ax^3+bx^2+cx+d. (a is not
necessarily 9). Now f
Ô(x) = 3ax^2+2bx+c must be 0 when x = -2 (local max) or when
x=1 (local
min), so f Ô(x) must factor as
f Ô(x) = 3ax^2+2bx+c = 3a ( x - 1 ) ( x + 2 )
Multiplying this out and solving for b and c, we find that
b = 3a/2 and c = -6a
So
f(x) = ax^3 + ( 3 / 2 )ax^2 - 6ax + d
Now we also have that
f(-2) = -8a + 6a + 12a + d = 10a + d = 3
and
f(1) = a + ( 3 / 2 )a - 6a + d = ( -7 / 2 )a + d = 0
Solving this for a and d gives us
a = 2 / 9 and d = 7 / 9
and therefore
b = 1 / 3 and c = -4 / 3
So the cubic you seek is
f(x) = ( 2 / 9 ) * x^3 + ( 1 / 3 ) * x^2 - ( 4 / 3 ) * x + 7
/ 9
--
submissions: post to k12.ed.math or e-mail to
k12math@k12groups.org
private e-mail to the k12.ed.math moderator:
kem-moderator@k12groups.org
newsgroup website: http://www.thinkspot.net/k12math/
newsgroup charter:
http://www.thinkspot.net/k12math/charter.html
===
Subject: Re: Please help
| Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
| and a local min value of 0 at 1.
|
|
|
f(x) = 27 x^2 + 2bx + c
f(x) = 54x + 2b
At local min or max, f(x) = 0.
Also, f(x) > 0 for min and < 0 for max.
--
submissions: post to k12.ed.math or e-mail to
k12math@k12groups.org
private e-mail to the k12.ed.math moderator:
kem-moderator@k12groups.org
newsgroup website: http://www.thinkspot.net/k12math/
newsgroup charter:
http://www.thinkspot.net/k12math/charter.html
===
Subject: Re: Please help
> Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
> and a local min value of 0 at 1.
f(x) = ax(x - 3)(x + 2)
--
submissions: post to k12.ed.math or e-mail to
k12math@k12groups.org
private e-mail to the k12.ed.math moderator:
kem-moderator@k12groups.org
newsgroup website: http://www.thinkspot.net/k12math/
newsgroup charter:
http://www.thinkspot.net/k12math/charter.html
===
Subject: Re: Please help
> Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
> and a local min value of 0 at 1.
f(x) = ax(x + 2)(x - 3)
===
Subject: Re: Please help
>>Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
>>and a local min value of 0 at 1.
> f(x) = ax(x + 2)(x - 3)
Why should
(1) the derivative of a cubic be a cubic?
(2) the derivative have zeros at 0 and 3?
Just curious.
Why I ask:
(1) A polynomial of degree n has a polynomial
of degree n-1 as its derivative.
(2) Generally, one looks for local extrema to
occur at zeros of the derivative.
Dale.
Dale
===
Subject: Re: Please help
>>Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at
-2
>>and a local min value of 0 at 1.
> f(x) = ax(x + 2)(x - 3)
> Why should
> (1) the derivative of a cubic be a cubic?
> (2) the derivative have zeros at 0 and 3?
f(x) = a(x + 2)(x - 1)
===
Subject: householder matrices
I am a bit confused by the definition of a 
Ôhouseholder
matrix. According
to the texts I have come across , the householder matrix is
defined as :-
H = I - 2 x x^T
Where I is the identity matrix, and Ôx a 
vector. What I dont
understand
is that the term Ô2 x x^T is a scalar and not a 
matrix, so
how can you
subtract a scalar from a matrix??
Also does the definition of the householder matrix extend to
complex
matrices? so that the definition maybe has the analogue :-
H = I - 2 x x^H
Where ÔH is the Hermittian. Would that 
expression hold for
the complex
case?
===
Subject: Re: householder matrices
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLEbZa12134;
>I am a bit confused by the definition of a 
Ôhouseholder
matrix.
According
>to the texts I have come across , the householder matrix is
defined
as :-
>H = I - 2 x x^T
>Where I is the identity matrix, and Ôx a 
vector. What I dont
understand
>is that the term Ô2 x x^T is a scalar and not 
a matrix, so
how can
you
>subtract a scalar from a matrix??
>Also does the definition of the householder matrix extend to
complex
>matrices? so that the definition maybe has the analogue :-
>H = I - 2 x x^H
>Where ÔH is the Hermittian. Would that 
expression hold for
the
complex
>case?
Jeremy,
Think of x as a n by 1 matrix. Then x^T x is a 1 by 1 matrix,
i.e.
a scalar, and x x^T is an n by n matrix where the (i,j)
element is
(x_i)(x_j). Clearly, x x^T is symmetric.
Yes, there is a Hermitian generalization just as you suspect.
Householder matrices are used in the QR decomposition of a
matrix.
Each Householder transformation is unitary (or simply
orthogonal
in the real case) as can be easily verified by taking h h^H.
(Here is am using lower case Ôh for the 
Householder matrix
so as
not to confuse it with the ÔH for Hermittian.) 
There is a
very
simply algorithm to chose n Householder matrices such that any
matrix A can be decomposed into A = QR where Q is the unitary
matrix resulting from the product of the Householder matrices
and R is upper triangular. See
http://planetmath.org/encyclopedia/
HouseholderTransformation.html
- MO
===
Subject: Re: householder matrices
Jeremy Watts <jwatts1970@hotmail.com> dixit:
>I am a bit confused by the definition of a 
Ôhouseholder
matrix.
According
>to the texts I have come across , the householder matrix is
defined as :-
>H = I - 2 x x^T
>Where I is the identity matrix, and Ôx a 
vector. What I
dont understand
>is that the term Ô2 x x^T is a scalar and not 
a matrix, so
how can you
>subtract a scalar from a matrix??
x is a n-dimensional unit column vector, so x^T, which means
transpose
of x, is a row vector. When you multiply, you get an n x n
matrix
which consists of dot products, not a scalar. So this formula
is
correct, in fact its the formula for a reßection across 
the
line
which has x as a unit direction vector. Thats what a
Householder
matrix is, if I remember correctly. (Its easy to derive 
this
formula
if you make a drawing in 3d, for example, using vectors).
>Also does the definition of the householder matrix extend to
complex
>matrices? so that the definition maybe has the analogue :-
>H = I - 2 x x^H
>Where ÔH is the Hermittian. Would that 
expression hold for
the complex
>case?
I dont know but you could try for some complex vector x to
apply the
formula. Then matrix H must be a reßection, so it must have
the
properties of a reßection like det(H)=-1 and H^2 = Identity,
and it
must be orthogonal.
===
Subject: Re: householder matrices
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLG9Fw21486;
>Jeremy Watts <jwatts1970@hotmail.com> dixit:
>>I am a bit confused by the definition of a 
Ôhouseholder
matrix.
According
>>to the texts I have come across , the householder matrix is
defined
as :-
>>H = I - 2 x x^T
>>Where I is the identity matrix, and Ôx a 
vector. What I
dont
understand
>>is that the term Ô2 x x^T is a scalar and not 
a matrix, so
how can
you
>>subtract a scalar from a matrix??
>x is a n-dimensional unit column vector, so x^T, which means
transpose
>of x, is a row vector. When you multiply, you get an n x n
matrix
>which consists of dot products, not a scalar. So this
formula is
>correct, in fact its the formula for a reßection across 
the
line
>which has x as a unit direction vector. Thats what a
Householder
>matrix is, if I remember correctly. (Its easy to derive 
this
formula
>if you make a drawing in 3d, for example, using vectors).
>>Also does the definition of the householder matrix extend to
complex
>>matrices? so that the definition maybe has the analogue :-
>>H = I - 2 x x^H
>>Where ÔH is the Hermittian. Would that 
expression hold for
the
complex
>>case?
>I dont know but you could try for some complex vector x to
apply the
>formula. Then matrix H must be a reßection, so it must have
the
>properties of a reßection like det(H)=-1 and H^2 = Identity,
and it
>must be orthogonal.
h = I - 2 x x^H is unitary, i.e. (h)(h^H) = (h^H)(h) = I.
h^2 will not necessarily equal I.
- MO
===
Subject: testing
just testing
===
Subject: testing
just testing
===
Subject: Book To Do and Learn Math Equations?
Can anyone recommend a math textbook with problems I can
practice on? I am
wanting to learn more on the equations as they pertain to
physics (for
example, string theory, etc...).....I see these equations...I
just want to
know how to read and solve them.
Would differential equations be a good place to start?
BTW, I do have some math background (calculus, diff equations,
physics....about 12 years ago).
===
Subject: Re: Book To Do and Learn Math Equations?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLFfMK18704;
Anthony <Anthony3@comcast.net>
http://mathforum.org/epigone/alt.math.undergrad/lunswonggen
> Can anyone recommend a math textbook with problems I can
> practice on? I am wanting to learn more on the equations
> as they pertain to physics (for example, string theory,
etc...)
> .....I see these equations...I just want to know how to
> read and solve them.
> Would differential equations be a good place to start?
> BTW, I do have some math background (calculus, diff
equations,
> physics....about 12 years ago).
Id suggest getting one of those texts for a two semester
advanced undergraduate level math for engineers and physicists
type course -- mostly to review and fill in your gaps (see
[1]),
and then when youre finished with it, begin 
working through
one of the texts used for a two semester beginning graduate
level mathematical methods course (see [2]).
[1] Mathematical Methods in the Physical Sciences
by Mary L. Boas
Advanced Engineering Mathematics
by Erwin Kreyszig (a solutions manual is available)
[2] Mathematical Methods of Physics
by Jon Mathews and Robert L. Walker
(this is probably more mid-way between
[1] and [2] than it is [2])
Mathematical Methods for Physicists
by George B. Arfken, Hans J. Weber, Hans-Jurgen Weber
Methods of Theoretical Physics
by Philip M. Morse and Herman Feshbach
Incidentally, a nice text that focuses a lot on
asymptotic analysis in a way that is both applied
and quite mathematically rigorous is
Advanced Mathematical Methods for Scientists and Engineers
by Carl Bender and Steven Orszag
[1] and [2] wont get you anywhere near where what 
youre
looking for, but itll give you the foundation to
(co-currently with [2] would be best) go through the
standard graduate level stuff in physics that everything
later rests on, namely classical mechanics (Goldsteins 
book),
classical electrodynamics (Jacksons book), and quantum
mechanics (Merzbacher, Schiff, Messiah).
By this point (if not well before), you should have enough
background, along with knowing your own strengths and
interests, to know what you want to begin specializing in.
For something like string theory, youre going to need
quite a bit of pure mathematics along the lines of topology,
functional analysis, differential geometry, etc. But unless
youre primarily interested in mathematics, 
youd probably
be best served by going through [1] and [2] first, especially
if youre doing this on your own.
Dave L. Renfro
===
Subject: Re: JSH: Did you ever finish your FLT proof?
> May I know in what message JSH talks about FLT?
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: need a ruler? go to my free online/onscreen
ruler project
>
> Did you ever need an online ruler while you were sitting in
front of
> your computer? now you can come to this page and use this
website to
> measure things.
> http://home.earthlink.net/~moon8500/davidmoon.html
On the other hand, if I want to measure accurately, Ill get
a real
ruler. Yours is not accurate on my monitor.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Empty SET
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLG2cg21109;
Hi everyone,
I want to ask about a partition of a set.
we know for a set of set to be a partition for A there is 3
conditions
must satisfy:
1-Empty set isnt belong to the partition i.e.none of the 
set
of the
partition must be the empty set.
why does this condition is important ???
can anyone explain to me ..
===
Subject: Re: Empty SET
> Hi everyone,
> I want to ask about a partition of a set.
> we know for a set of set to be a partition for A there is 3
conditions
> must satisfy:
> 1-Empty set isnt belong to the partition i.e.none of the
set of the
> partition must be the empty set.
> why does this condition is important ???
> can anyone explain to me ..
In some elementary situations it mightnt do much harm to
include
the empty set. However, it causes trouble if you need to
_count_ the
equivalence classes, e.g. in the proof of Lagranges Theorem
in group
theory. The empty class would increase the count by one,
messing up
such properties.
Ken Pledger.
===
Subject: Re: Pells eq. algorythms
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBLG9Fb21490;
(oops)!=all the primes of X^2+5*Y^2 of form 4*k+3
>Hi Todd,
>I gave -up on showing my method for d=61.
>That is because I want to simplify my presentation of the
method for
>which I have a proof that It works allways.I presented in
writing to
>Mr.Prof K.Sound from University Of Michigan and some of it
before
that
>to another professors.
>You are going to see what Euler miss to find.
>I never seen any where this :in no elemenetary theory of
numbers
books
>or advanced ones.Well ,it is only elementary high school
algebra!
>Here is a simple exemple:
> Lets choose D=29.
> Lets choose a<(29)^(1/2)>a+1
> a=5
> Now we write:
> 5^2-29=(1)*(-4)
> b^2-29=-4*k
>We choose b as being the largest number <(29)^(1/2)such that
> b+5 is divisible by 4 and we continuu the algorithm
following this
>rule.
>so we have:
> 5^2-29=(1)*(-4) I
> 3^2-29=-4*5 II
> 2^2-29=-5*5 III
> 3^2-29=-5*4
> 5^2-29=-4*1
>Now we know that
(a^2+s*b^2)*(c^2+s*d^2)=(a*c-s*b*d)^2+s*(a*d+b*c)^2
>So we multiply I*II=a^2-29*b^2=5*(4)^2
> The proof shows that 4^2 simplifies and we get;
> I=c^2-29*d^2=5
> Now we multiply (I)*III= e^2-29*f^2=-5*5^2
>After we divide by 5^2 we continuu this algorithm to the end
and get
> x^2-29*y^2=-1
>Now we multiply this identity by itself and get
> X^2-29*Y^2=1
> The proof is elementary high school algebra technique.
>How in the world Euler miss to find this algorythm and he
never was
>able to proof Pells equation.
>More than that.It took for him 7 years to proof that a
prime=4*k+1
>can be represented as a^2+b^2 and to get a method how to find
this
>representation.Whereas this algorithm which i called 
Fermats
>Algorythm shows how to do it:
> As you see we got above the solution
> x^2+1=29*y^2
> We can get from above
> a^2+1=29*b where a <29 and apply Fermats algorythm
> c^2+1=b*e where c+a=b*v
> .........
> .........
> x^2+1=y*1
> Now we do the multiplication and get the representation
> X^2+Y^2=29
> We y*1 as the last product.
> Well is needed only high school algebra to show that if we
have
> x^2+s=D*b
> then in the last product y*v we have 1<v<(s+1)/2
>In this way Fermat proved all his statements about the
representations
>of diferrent primes .I proved them too.
>Try the one about that all the primes of
> X^2+5*Y^2 have as the last digit 3 or 7
>I am not beating the bush aroun in this letter(oops!)
>
>Hi Todd,
>Well, I apreciate very much your answer.
>>Sure. Well, it was my pleasure.
>I observed the diference in the notation : I used A and you
use D
>>Makes no difference.
>I think this is coming from the the books we studied.
>>Well, my solution comes from my own playing around with the
>>problem, not from books. Although I have read a few in my
life :)
>Did you ever tried or see the cyclic method used in India
back in
>>800
>A.D to find the solutions?
>>No, but it sounds interesting. Why dont you tell us about
it?
>>The word cyclic reminds me of the fact that the integers
which
>>pop out of the continued fraction method are palindromic
(e.g.
>>2, 1, 1, 1, 2 in my example).
>Did you wonder If Archimedes had a method to solve the eq.
when in
>theCatlle
>story- problem he get us to resolve a Pells equation at 
the
end
of
>calculations?
>>I seem to recall reading something about this in the
American
>>Mathematical Monthly some years ago. I recall the discussion
>>was very interesting and maybe Ill dig it up and reread
it, now
>>that you bring it up.
>Did you ever wonder what was Fermats method to resolve it?
>>Well, I havent studied Fermat. I have the impression that
he
>>used his method of descent to solve a great many problems,
and
>>I half-expect youll be telling me something more about
this.
>Well , I know,in this Age we are conditioned to be very
pragmatic,
>learn to use a method well and be able to Aply it.So less
and less
>>we
>dont have time to wonder
>or find diferent ways to resolve a problem.
>>Use of words like conditioned, or imputations that people
>>(like me, presumably) dont wonder any more, are rather
obnoxious
>>and presumptuous, dont you think? If you want to have a
discussion,
>>then you should really be more polite, my friend.
>I think you are very nice for writing down for me the use of
>continuu
>fractions in this
> case ,but I was looking beyond this when I put the
question.I
>mean,I
>wanted to hear some original coming from Wondering.
>>down something I knew, thinking it was a real question (and
not
>>some sort of challenge, which is how it looks to me now).
Ill
>>try to be more careful next time.
>Yes, You gave me the case D=61.In one day,two,three days
from now I
>can show to you my algorythm which I call Fermats
algorythm.Well
>,this algorythm is much simple than what your preference(my
>opinion).This algorythm shows how Fermat used it to proof his
>staments about representations of different primes too.
>In another words it opens a Window in the Hystorie of past
>methods
>>Well, great! Im always pleased to learn new things. I do
>>think the method I gave is pretty simple, and I suspect
that at
>>some level other solutions are in some sense equivalent,
but Ill
>>wait to hear more from you.
>Remember, Euler was a master of Algorithms and he fail to
find one
>>for
>Pellequation
>Some Professors know about this method because I PUT IT OUT
,I have
>shown it to
> them in writting.But you have to show to me that you look a
little
>bit in the Hyistory of mathematics.Take a course If you did
not or
>studied it by yourself.
>It is fascinated and insight inspiring.That if you like
mathematics
>>Well, I congratulate you on your successes with Professors,
but...
>>I have to show you *what*?
>>I congratulate you further -- youve got quite a lot of
nerve.
>>Todd Trimble
===
Subject: Re: Pells eq. algorythms
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iBKMlYj30260;
Hi Todd,
I am sorry.It is possible that My humor is of bad taste to you
II was beating the bush in mathematicsliketerms
I will stop it.
made the
observation
I apologise again
george ghiata
>>Hi Todd,
>>Well, I apreciate very much your answer.
>Sure. Well, it was my pleasure.
>>I observed the diference in the notation : I used A and you
use D
>Makes no difference.
>>I think this is coming from the the books we studied.
>Well, my solution comes from my own playing around with the
>problem, not from books. Although I have read a few in my
life :)
>>Did you ever tried or see the cyclic method used in India
back in
>800
>>A.D to find the solutions?
>No, but it sounds interesting. Why dont you tell us about
it?
>The word cyclic reminds me of the fact that the integers
which
>pop out of the continued fraction method are palindromic
(e.g.
>2, 1, 1, 1, 2 in my example).
>>Did you wonder If Archimedes had a method to solve the eq.
when in
>>theCatlle
>>story- problem he get us to resolve a Pells equation at
the end of
>>calculations?
>I seem to recall reading something about this in the American
>Mathematical Monthly some years ago. I recall the discussion
>was very interesting and maybe Ill dig it up and reread 
it,
now
>that you bring it up.
>>Did you ever wonder what was Fermats method to resolve 
it?
>Well, I havent studied Fermat. I have the impression that 
he
>used his method of descent to solve a great many problems,
and
>I half-expect youll be telling me something more about 
this.
>>Well , I know,in this Age we are conditioned to be very
pragmatic,
>>learn to use a method well and be able to Aply it.So less
and less
>>dont have time to wonder
>>or find diferent ways to resolve a problem.
>Use of words like conditioned, or imputations that people
>(like me, presumably) dont wonder any more, are rather
obnoxious
>and presumptuous, dont you think? If you want to have a
discussion,
>then you should really be more polite, my friend.
>>I think you are very nice for writing down for me the use of
continuu
>>fractions in this
>> case ,but I was looking beyond this when I put the
question.I
mean,I
>>wanted to hear some original coming from Wondering.
>down something I knew, thinking it was a real question (and
not
>some sort of challenge, which is how it looks to me now).
Ill
>try to be more careful next time.
>>Yes, You gave me the case D=61.In one day,two,three days
from now I
>>can show to you my algorythm which I call Fermats
algorythm.Well
>>,this algorythm is much simple than what your preference(my
>>opinion).This algorythm shows how Fermat used it to proof
his
>>staments about representations of different primes too.
>>In another words it opens a Window in the Hystorie of past
methods
>Well, great! Im always pleased to learn new things. I do
>think the method I gave is pretty simple, and I suspect that
at
>some level other solutions are in some sense equivalent, but
Ill
>wait to hear more from you.
>>Remember, Euler was a master of Algorithms and he fail to
find one
>for
>>Pellequation
>>Some Professors know about this method because I PUT IT OUT
,I have
>>shown it to
>> them in writting.But you have to show to me that you look
a little
>>bit in the Hyistory of mathematics.Take a course If you did
not or
>>studied it by yourself.
>>It is fascinated and insight inspiring.That if you like
mathematics
.
>Well, I congratulate you on your successes with Professors,
but...
>I have to show you *what*?
>I congratulate you further -- youve got quite a lot of
nerve.
>Todd Trimble
===
Subject: Intergrate[acrtan x] dx
Hi!
I am trying to integrate arctan x, but doesnt get the right
answer. Here is
what I am doing:
I[ arctan x ] dx = I[ 1 * arctan x ] dx
substitute u = arctan x, v = 1
x* arctan x - I[ 1/(1+x^2) * x]dx
the latter fraction is becoming x/(1+x^2), how do I integrate
this one?
Maybe I am missing something here..
--
===
Subject: Re: Intergrate[acrtan x] dx
>Hi!
>I am trying to integrate arctan x, but doesnt get the 
right
answer. Here
is
>what I am doing:
>I[ arctan x ] dx = I[ 1 * arctan x ] dx
>substitute u = arctan x, v = 1
>x* arctan x - I[ 1/(1+x^2) * x]dx
>the latter fraction is becoming x/(1+x^2), how do I
integrate this one?
>Maybe I am missing something here..
You could let z = x^2, so that dz = 2x dx.
Brian
===
Subject: Re: Intergrate[acrtan x] dx
>>Hi!
>>I am trying to integrate arctan x, but doesnt get the
right answer. Here
is
>>what I am doing:
>>I[ arctan x ] dx = I[ 1 * arctan x ] dx
>>substitute u = arctan x, v = 1
>>x* arctan x - I[ 1/(1+x^2) * x]dx
>>the latter fraction is becoming x/(1+x^2), how do I
integrate this one?
>>Maybe I am missing something here..
>You could let z = x^2, so that dz = 2x dx.
>Brian
Better yet, let z = 1 + x^2 and you get dz = 2x dx.
Brian
===
Subject: Re: Intergrate[acrtan x] dx
>Hi!
>I am trying to integrate arctan x, but doesnt get the 
right
answer. Here
>is
>what I am doing:
>I[ arctan x ] dx = I[ 1 * arctan x ] dx
>substitute u = arctan x, v = 1
>x* arctan x - I[ 1/(1+x^2) * x]dx
>the latter fraction is becoming x/(1+x^2), how do I
integrate this one?
>Maybe I am missing something here..
>>You could let z = x^2, so that dz = 2x dx.
>>Brian
> Better yet, let z = 1 + x^2 and you get dz = 2x dx.
> Brian
--
Ronny M
===
Subject: Re: computing the logarithm of arbitrary base
>> hey folks, silly question from someone who prolly never
should have
embarked
>> on a 3 year bachelors degree in mathematics... anyway, i
would like to
>> compute the b-base logarithm of a number, but my
calculator seems only
allow
>> me the bases e and 10... hints anyone??
> Learn mathematics, ditch the stupid calculator.
> Have any of them ever passed a math class?
> Stick with them and youll be doing the same.
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Well said. Students are getting too dependent on technology
for
EVERYTHING and are getting the wrong idea about what
mathematics is.
Aristotle Polonium
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
===
Subject: Re: computing the logarithm of arbitrary base
>hey folks, silly question from someone who prolly never
should have
embarked
>on a 3 year bachelors degree in mathematics... anyway, i
would like to
>compute the b-base logarithm of a number, but my calculator
seems only
allow
>me the bases e and 10... hints anyone??
log_b(x) = log_10(x) / log_10(b)
http://oakroadsystems.com/math/loglaws.htm#NewBase
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
Dont move, or Ill fill you full 
of [... pause ...] little
yellow bolts of light. -- Farscape, first episode
===
Subject: Re: Is zero even or odd?
> How many universes are in a black hole ?
> Oh, this sounds like even more fun. Something we know even
less
> about ...
> I would say about a black-holes-worth.
Well, if theres at least one universe inside a black hole,
then that
universe could contain another black hole, and so forth.
Because of the Schwarzschild radius, the is at least one
universe inside
the black hole separate from ours.
By induction, the answer is: infinitely many.
Michael
--
Still an attentive ear he lent Her speech hath caused this
pain
But could not fathom what she meant Easier I count it to
explain
She was not deep, nor eloquent. The jargon of the howling main
-- from Lewis Carroll: The Three Usenet Trolls
===
Subject: Re: Is zero even or odd?
> 0 cant be divided by itself,
>> Sure it can: 0 / 0 = 0 * (1 / 0) = 0 * infinity = 1
>> It works if the only three numbers in the universe are
>> 0, 1, and infinity -- A number system that seems very
>> suited to usenet.
>Except for the fact that: 0 / 0 = undefined
>Or actually more correct: n / 0 = undefined
>> The two are not the same.
>> The definition of the ratio a/b is
>> a/b = r iff b*r = a
>> for the case of n/0 there is no r such that r*0 = n
(follows from the
>> definition of zero. Therefore n/0 (for non zero n) *does
not exist*.
>> On the other hand, for 0/0, every r qualifies since for
every r, r*0 =
>> 0 (the definition of zero, again). Therefore, 0/0 is truly
undefined,
>> in the sense that it is impossible to *uniquely* assign a
value to the
>> ratio r.
>> Mati Meron | When you argue with a fool,
>> meron@cars.uchicago.edu | chances are he is doing just the
same
>It depends on how you get there, [sin(x)]/x is certainly
defined for all
>values of x including 0 and infinity.
Thats a different thing. Here youre talking 
not about a
plain value
but a limit (of an infinite set of values). And this depends
how you
get there. Thus, [sin(0)]/0 is undefined. On the other hand,
lim_x->0 {[sin[x]/x} is defined and equal to 1.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the
same
===
Subject: Re: Is zero even or odd?
> 0/0 is clearly, if anything, a constant expression. And it
turns out
> [to some] that its value is undefined.
> Better minds than can be found here have argued this and not
reached
> any conclusion. ÔUndefined is 
the answer given by the
teacher in
the
> 7th grade, and will serve for all practical purposes.
> Maybe what is needed is a New Number = Ô* (or 
something) =
Any
Number You
> Want.
> Just FYI, if youre performing arithmetic using the IEEE 
754
standard,
I doubt if there are any mathematicians who care a hoot about
definitions made by engineers for computational convenience
Franz
===
Subject: Re: Is zero even or odd?
0/0 is clearly, if anything, a constant expression. And it
> turns out
> [to some] that its value is undefined.
Better minds than can be found here have argued this and not
> reached
> any conclusion. ÔUndefined is 
the answer given by the
teacher in
> the
> 7th grade, and will serve for all practical purposes.
Maybe what is needed is a New Number = Ô* (or 
something) =
Any
> Number You
> Want.
> Just FYI, if youre performing arithmetic using the IEEE 
754
> standard,
> I doubt if there are any mathematicians who care a hoot
about
> definitions made by engineers for computational convenience
Well, then, how about yall stop crossposting from Hell to
breakfast?
Followups set.
Xho
--
-------------------- http://NewsReader.Com/
--------------------
Usenet Newsgroup Service $9.95/Month 30GB
===
Subject: Re: Is zero even or odd?
> 0 cant be divided by itself,
Sure it can: 0 / 0 = 0 * (1 / 0) = 0 * infinity = 1
It works if the only three numbers in the universe are
> 0, 1, and infinity -- A number system that seems very
> suited to usenet.
>>Except for the fact that: 0 / 0 = undefined
>>Or actually more correct: n / 0 = undefined
> The two are not the same.
> The definition of the ratio a/b is
> a/b = r iff b*r = a
> for the case of n/0 there is no r such that r*0 = n
(follows from
the
> definition of zero. Therefore n/0 (for non zero n) *does not
exist*.
> On the other hand, for 0/0, every r qualifies since for
every r,
r*0 =
> 0 (the definition of zero, again). Therefore, 0/0 is truly
undefined,
> in the sense that it is impossible to *uniquely* assign a
value to
the
> ratio r.
> Mati Meron | When you argue with a fool,
> meron@cars.uchicago.edu | chances are he is doing just
the same
> It depends on how you get there, [sin(x)]/x is certainly
defined
for all
> values of x including 0 and infinity.
If you knew any maths worth talking about, you would have
known that
sin(0) / 0 is not the same as the limit of sin(x) / x as x
tends to 0.
The first is undefined and the second is unity.
Now it is your turn: What do you know about sin (infinity) /
infinity
?
Franz
===
Subject: Re: Is zero even or odd?
> 0 cant be divided by itself,
>> Sure it can: 0 / 0 = 0 * (1 / 0) = 0 * infinity = 1
>> It works if the only three numbers in the universe are
>> 0, 1, and infinity -- A number system that seems very
>> suited to usenet.
Except for the fact that: 0 / 0 = undefined
Or actually more correct: n / 0 = undefined
> The two are not the same.
>> The definition of the ratio a/b is
>> a/b = r iff b*r = a
>> for the case of n/0 there is no r such that r*0 = n
(follows from
> the
>> definition of zero. Therefore n/0 (for non zero n) *does 
not
> exist*.
>> On the other hand, for 0/0, every r qualifies since for
every r,
> r*0 =
>> 0 (the definition of zero, again). Therefore, 0/0 is truly
> undefined,
>> in the sense that it is impossible to *uniquely* assign a
value to
> the
>> ratio r.
>> Mati Meron | When you argue with a fool,
>> meron@cars.uchicago.edu | chances are he is doing just
> the same
>> It depends on how you get there, [sin(x)]/x is certainly
defined
> for all
>> values of x including 0 and infinity.
> If you knew any maths worth talking about, you would have
known that
> sin(0) / 0 is not the same as the limit of sin(x) / x as x
tends to 0.
> The first is undefined and the second is unity.
Tell that to all the book publishers who print curves for
sinx/x.
> Now it is your turn: What do you know about sin (infinity) /
infinity
> Franz
No problem. Sin x is bounded between +/- 1 for all values of
x. A finite
number divided by infinity is 0.
Tam
===
Subject: Re: Is zero even or odd?
) <snip>
)> If you knew any maths worth talking about, you would have
known that
)> sin(0) / 0 is not the same as the limit of sin(x) / x as x
tends to 0.
)> The first is undefined and the second is 
unity.
) Tell that to all the book publishers who print curves for
sinx/x.
If you zoom in on those printed curves far enough, youll
notice that
there is no ink at the actual point (0,1).
SaSW, Willem
--
Disclaimer: I am in no way responsible for any of the
statements
made in the above text. For all I know I might be
drugged or something..
No Im not paranoid. You all think Im 
paranoid, dont you !
#EOT
===
Subject: Re: Is zero even or odd?
>> I know 0 is neither negative or positive but what about
odd/even? I
> think
>> its even.
> It is essentially neutral for it signifies a number that
isnt there
> because it lacks quantity. It is the symbol for nothingness
therefore
> it isnt odd nor even. For example, take the addition of
all numbers
> from -infinity to +infinity, what do you
> get?...zero!...nothin!....nada! The sum of everything is
equal to
> nothing!
> In physics, the conservation of energy states that no
energy can be
> created or destroyed. Therefore, if you add up all of the
negative
> potential energy in the universe with the positive kinetic
energy, you
> get NOTHING!....hence the total energy in the universe is
zero.
> Therefore, we are all essentially made up of nothing.
> Sleep tight!
If you have two apples and you eat the two apples there are
now no apples
in the whole of an infinite Universe? Zero? What of the term
ground
zero?
Does it mean no ground? What of quality, or is it quantity,
infinite
zero?
Zero point energy? Running a zero balance? Hawking time zero
on the
clock? My own four dimensions of time in which one dimension
is zero
(history (pasts-futures), frequency, relativity (gain in and
loss of), and
zero)? Zeroing in on? And, most particularly, base2, base4,
base8,
base16.....?
Zero can be an anchor.
mathematicians in the early twentieth century conceived
monstrous-seeming
objects made by the technique of adding or removing infinitely
many parts.
One such shape is the Sierpinski carpet, constructed by
cutting the center
one-ninth of a square; then cutting out the centers of the
eight smaller
squares that remain; and so on. The three-dimensional
analogue is the Menger
sponge, a solid-looking lattice that has an infinite surface
area, yet zero
volume.
Brad