A few questions

Agnishom · 2012-12-07 21:53:06

I see

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?

bobbym · 2012-12-07 22:07:31

Hi Agnishom;

When you get some time go here.

http://betterexplained.com/articles/fun … rithmetic/

http://www.math.okstate.edu/~wrightd/cr … ode12.html

Have patience and do not expect to understand it all at once.

Agnishom · 2012-12-08 02:23:05

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

bobbym · 2012-12-08 08:14:03

Hi Agnishom;

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

noelevans · 2012-12-08 08:33:37

Hi phrontister!

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

bobbym · 2012-12-08 09:34:32

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.

Agnishom · 2012-12-08 17:04:39

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

noelevans · 2012-12-09 03:49:56

Answer:

Agnishom · 2012-12-09 03:58:04

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

scientia · 2012-12-09 22:30:54

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)

scientia · 2012-12-09 22:53:10

Sorry, I made a mistake.

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

bobbym · 2012-12-09 22:55:55

Hi scientia;

scientia · 2012-12-10 00:25:02

Yeah, as I said I made a mistake.

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?

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

bobbym · 2012-12-10 09:03:09

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.

Agnishom · 2012-12-10 15:18:25

What are these x,y and z?

noelevans · 2012-12-11 00:28:28

Hi Agnishom!

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!

Agnishom · 2012-12-11 02:15:48

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

bobbym · 2012-12-11 02:18:00

Hi Agnishom;

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

Agnishom · 2012-12-11 02:52:46

Then How did you work them out?

bobbym · 2012-12-11 02:57:58

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

Agnishom · 2012-12-11 04:05:28

Do you have the proof?

bobbym · 2012-12-11 04:09:32

Hi;

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

Agnishom · 2012-12-11 04:13:19

I just can not post the proof to that.

Why? Whats wrong?

bobbym · 2012-12-11 04:16:01

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.

Agnishom · 2012-12-11 04:18:20

Please show me

Math Is Fun Forum

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

Re: A few questions

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

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

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

Re: A few questions

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

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

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

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

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

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

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

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

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

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