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#76 2012-12-07 21:53:06

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
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Re: A few questions

I see smile

bobbym wrote:

This way is easier to understand at first but I wanted to show you how it was done with mods first. For these MO problems a very good understanding of the properties of mods is required.

Any article or anything you would suggest me to read for clearing my idea on mods?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#77 2012-12-07 22:07:31

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#78 2012-12-08 02:23:05

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
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Re: A few questions

Would you suggest any good book that is freely available for Number Theory?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#79 2012-12-08 08:14:03

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi Agnishom;

I will need to think about it. I will post one here.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#80 2012-12-08 08:33:37

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Hi phrontister!

I had never noticed those symbols!  Thanks for pointing them out.  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#81 2012-12-08 09:34:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi Agnishom;

Look for this book it was my first number theory book.

"Recreations in the Theory of Numbers" by
Albert H. Beiler

It will be tough in some spots so do not be afraid to jump around. Since you have programming skills you might want to solve some of the problems in that way.

It stresses a problem solving approach. It is not exactly an introductory text but I do not believe there is one and I have looked at hundreds.

Number theory is the mathematicians favorite and they are not willing to give up its secrets without a fight.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#82 2012-12-08 17:04:39

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Thanks! I will try to find it smile
Question 8: Find all integer sided triangles whose area and peremeter are equal


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#83 2012-12-09 03:49:56

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Answer:

wave


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#84 2012-12-09 03:58:04

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Excuse me noelevans, I think we had been talking about Triangles


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#85 2012-12-09 22:30:54

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: A few questions

Let the sides be
. The perimeter is
and the area is
by Heron's formula.

Hence

Setting

:

Since

are sides of a triangle, we can make the substitution
,
,
, which gives



Thus

are integers, as
are integers; they are also positive as the sum of two sides of a triangle must be strictly greater than the third side. Substituting in the above equation:

So the RHS is divisible by 4. This means either (i) two of

equal 2, or (ii) one of
equals 4. In case (i) we have (taking
WLOG)
, an absurdity. So case (ii) must hold. WLOG let
. Then

Thus

must be an integer
or
, which gives
or
.

Substituting back gives

or
. Hence the only triangle with equal area and perimeter is the one with sides 6, 8, 10.

Last edited by scientia (2012-12-09 22:58:27)

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#86 2012-12-09 22:53:10

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: A few questions

Sorry, I made a mistake.




This means that either
are integers or of the form
for
. There may be other answers. neutral

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#87 2012-12-09 22:55:55

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi scientia;


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#88 2012-12-10 00:25:02

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: A few questions

Yeah, as I said I made a mistake. sad




Let
,
,
, where
are positive integers (so
themselves are not necessarily integers). This gives

which is what is required to be solved.

I haven't gone very far, have I? rolleyes

Last edited by scientia (2012-12-10 00:25:49)

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#89 2012-12-10 09:03:09

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi scientia;

I wasn't correcting I was just hoping that you could make a simple adjustment to your way if you saw more data. I do not even know yet if those are the only possible solutions.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#90 2012-12-10 15:18:25

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

What are these x,y and z?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#91 2012-12-11 00:28:28

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Hi Agnishom! smile

Boy! Did I misread the problem or what?  Sure was easy for the square, but a horse of a different
color for the triangles.  I think you will find that the only two Pythagorean triples that will work are
the first two that bobbym  listed.  The basic Pythagorean triples are of the form

                                           (u²+v², u²-v², 2uv) with 0<v<u.

(6,8,10) is a scalar multiple of (3,4,5); that is, (2*3,2*4,2*5) where u=2 and v=1.
(5,12,13) is a basic Pythagorean triple where u=3 and v=2.

I would have responded sooner, but my internet connection has been out.

Have a great day!


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#92 2012-12-11 02:15:48

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Excuse me noelevans, What if the triangle isn't right angled?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#93 2012-12-11 02:18:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi Agnishom;

There are 5 solutions, I just can not post the proof to that.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#94 2012-12-11 02:52:46

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Then How did you work them out? dunno


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#95 2012-12-11 02:57:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

The proof supplied the 5 answers. I checked them using geogebra and that was not easy either.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#96 2012-12-11 04:05:28

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Do you have the proof?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#97 2012-12-11 04:09:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

Hi;

Yes I do but it is not mine and the last part I do not understand well enough.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#98 2012-12-11 04:13:19

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

I just can not post the proof to that.

Why? Whats wrong?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#99 2012-12-11 04:16:01

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: A few questions

For one thing, it could be wrong! If I do not understand it how can I explain it or be satisfied with it? Oh, he/she is right with the five solutions he gives. I checked them and they are correct. But supposing there are a couple of more solutions?

I will post it if you would like to see it.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#100 2012-12-11 04:18:20

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,885
Website

Re: A few questions

Please show me


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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