The Math Forum

For anyone with a love of math
 
HomeHome  CalendarCalendar  GalleryGallery  FAQFAQ  SearchSearch  MemberlistMemberlist  UsergroupsUsergroups  RegisterRegister  Log in  

Share | 
 

 All you could ever want from pythagoreas!!!

View previous topic View next topic Go down 
AuthorMessage
Dojo
Admin
Admin
avatar

Number of posts : 154
Location : Probably somewhere near a computer
Registration date : 2008-08-12

PostSubject: All you could ever want from pythagoreas!!!   Sun Aug 17, 2008 3:39 pm

Post pythagoreas proofs/integer triples you can find!!!!

_________________
~Dojo
Back to top Go down
View user profile http://themathforum.forumotion.com
mathblitz
Developer
Developer
avatar

Number of posts : 29
Age : 20
Location : GET AWAY STALKER
Registration date : 2008-08-16

PostSubject: Re: All you could ever want from pythagoreas!!!   Sun Aug 17, 2008 4:42 pm

heres some pythagoream triples:

3,4,5
5 12 13
7 24 25
9 40 41
11 60 61
13 84 85

see if you spot the pattern... (its easy i spotted it in third grade or fourth) Very Happy
Back to top Go down
View user profile
herefishyfishy1
Developer
Developer


Number of posts : 36
Age : 22
Location : An insignificant little blue-green planet 93 million miles from the sun Sol.
Registration date : 2008-08-16

PostSubject: Re: All you could ever want from pythagoreas!!!   Sun Aug 17, 2008 4:57 pm

You can assign an ordered pair to each of the triples. Call the pair (x,y). The first number in the triple is x2-y2. The second is 2xy. The third is x2+y2.
Back to top Go down
View user profile
AIME15
Hardcore TMF user
Hardcore TMF user
avatar

Number of posts : 163
Age : 20
Location : Pleasanton, CA
Registration date : 2008-08-13

PostSubject: Re: All you could ever want from pythagoreas!!!   Mon Aug 18, 2008 5:50 pm

This looks like special binomial products, like (a-b)^2, (a+b)^2, etc.

Also, at least one number in each triple is divisible by 4 and at least one is divisible by 5. I've never been able to prove it, although i know there is one.
Back to top Go down
View user profile
pythag011
School Math Nerd
School Math Nerd
avatar

Number of posts : 15
Registration date : 2008-08-17

PostSubject: Re: All you could ever want from pythagoreas!!!   Mon Aug 18, 2008 7:04 pm

AIME15 wrote:
This looks like special binomial products, like (a-b)^2, (a+b)^2, etc.

Also, at least one number in each triple is divisible by 4 and at least one is divisible by 5. I've never been able to prove it, although i know there is one.

The (x,y) thing actually generates all the pythagorean triples with all three numbers relatively prime. So then if x, y are both odd, then x^2-y^2 and x^2+y^2 and 2xy are all even and it's not relatively prime. So , when it is relatively prime, one of x and y must be even, so the 2xy is a multiple of 4.

For the divisibility by 5 thing, take the equation $x^2 + y^2 = z^2$ modulo 5, and there are no solutions if none of them are divisible by 5.
Back to top Go down
View user profile
AIME15
Hardcore TMF user
Hardcore TMF user
avatar

Number of posts : 163
Age : 20
Location : Pleasanton, CA
Registration date : 2008-08-13

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 1:14 pm

So you could do the same thing with 4?
Back to top Go down
View user profile
pythag011
School Math Nerd
School Math Nerd
avatar

Number of posts : 15
Registration date : 2008-08-17

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 2:57 pm

AIME15 wrote:
So you could do the same thing with 4?

Read the first sentence. Thouh I wrote it in a really ba d way...
Back to top Go down
View user profile
AIME15
Hardcore TMF user
Hardcore TMF user
avatar

Number of posts : 163
Age : 20
Location : Pleasanton, CA
Registration date : 2008-08-13

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 4:04 pm

Oh yeah, thanks.

Anyone find proofs of this yet?
Back to top Go down
View user profile
pythag011
School Math Nerd
School Math Nerd
avatar

Number of posts : 15
Registration date : 2008-08-17

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 4:28 pm

Here's a better proof of the fact that at least one element has to be a multiple of 4.

Take the equation x^2 + y^2 = z^2 in modulo 8.

A perfect square is either 1 ( if it is odd) (4 if it is even but not x is not a multiple of 4) or 0(if it is a multiple of 4) modulo 8.

If none of them are 0 modulo 8, Then x^2 + y^2 can either be 1+1 = 2, which is not a value a perfect square can have, 4+1=5(same thing) or 4+4 = 0, which implies z is a multiple of 4. Therfore, one of them has to be a multiple of 4.
Back to top Go down
View user profile
AIME15
Hardcore TMF user
Hardcore TMF user
avatar

Number of posts : 163
Age : 20
Location : Pleasanton, CA
Registration date : 2008-08-13

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 5:44 pm

i meant proofs ot eht ehorem, not the proofs of my statement earlier :DDD
Back to top Go down
View user profile
pythag011
School Math Nerd
School Math Nerd
avatar

Number of posts : 15
Registration date : 2008-08-17

PostSubject: Re: All you could ever want from pythagoreas!!!   Tue Aug 19, 2008 7:03 pm

The thereom that all pythagorean triples can be expressed as (c(x^2-y^2), 2xcy, and c(x^2+y^2))?

Here is a proof.

First, divide gcd(x,y,z) out. This will be c.

After that, one of (x,y) will be odd, and z will be odd. Why? If any two of them are even, all of them are even, and that can't happen 'cause we divide the gcd out. If z is even, then we consider the equation modulo 4, and since squares can only be 0 or 1 modulo 4, then x and y are both eve, contradiction.

Let's say y is even.

Let m be (x+z)/2, and let n be (z-x)/2. Then x= m-n and z = m+n. We get y the square of y is 4mn. If m and n are not relatively prime, then their sum and difference are also not relatively prime. Furthermore, y cannot be relatively prime because y^2 = z^2-x^2. So m and n are relatively prime. Since 4mn is a square and 4 is a square, mn must be a square. Because m and n are relatively prime, m must be a square and n must be a square. Now we get the thereom. QED
Back to top Go down
View user profile
ternary0210
School Math Nerd
School Math Nerd


Number of posts : 41
Age : 19
Registration date : 2008-08-17

PostSubject: Re: All you could ever want from pythagoreas!!!   Mon Sep 08, 2008 4:58 pm

15 112 113
16 30 34
32 60 68
64 120 136
17 144 145
19 180 181
21 220 221
Back to top Go down
View user profile
Sponsored content




PostSubject: Re: All you could ever want from pythagoreas!!!   

Back to top Go down
 
All you could ever want from pythagoreas!!!
View previous topic View next topic Back to top 
Page 1 of 1

Permissions in this forum:You cannot reply to topics in this forum
The Math Forum :: Middle School Topics :: Problem Solving :: Marathons-
Jump to: