Prove Pythagorean Triples: 3k, 4k, 5k | Answer Yay/Nay

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Discussion Overview

The discussion revolves around the claim that every Pythagorean triple can be expressed in the form 3k, 4k, 5k. Participants are examining the validity of this assertion and exploring definitions and examples related to Pythagorean triples.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that every Pythagorean triple can be expressed as 3k, 4k, 5k, using specific definitions for x, y, and z.
  • Another participant challenges this claim by providing a counterexample (5, 12, 13), indicating that it does not fit the proposed form.
  • Some participants note that the original problem may be incorrectly stated, as it does not account for known Pythagorean triples that do not conform to the 3k, 4k, 5k structure.
  • There is a suggestion that the problem might actually be asking to prove that triples of the form (3k, 4k, 5k) are indeed Pythagorean triples, which is a different assertion.
  • Concerns are raised about the implications of k not being an integer and how that affects the validity of the original claim.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of the claim that all Pythagorean triples can be expressed in the form 3k, 4k, 5k. Multiple competing views remain, with some asserting the claim is incorrect based on counterexamples.

Contextual Notes

There are unresolved questions regarding the exact wording of the original problem and its implications. The discussion highlights the need for clarity in definitions and the conditions under which the claim is made.

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Prove that every Pythagorean triple is of the form 3k, 4k, 5k. Could I say that 3k = x = 2st, 4k = y = t^2-s^2, and 5k = z = t^2 + s^2? those are the definitions of the pythagorean triple correct? can anyone say yea or nay? if nay, how can i make it correct?
 
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1+1=1 said:
Prove that every Pythagorean triple is of the form 3k, 4k, 5k.

You might want to check the exact wording of the problem, 5, 12, 13 is a Pythagorean triple that's not of the form you give.
 
the wording is correct.
 
He/she didn't say that k was necessarily an integer...

The general form of pythagorean triples is well known, try googling for them and I@m sure you'll find a nice proof.

What you wrote certainly isn't true as the counter example shows. One counter example disproves it, so how are you going to amend your question?

Where did the question appear?
 
1+1=1 said:
Prove that every Pythagorean triple is of the form 3k, 4k, 5k. Could I say that 3k = x = 2st, 4k = y = t^2-s^2, and 5k = z = t^2 + s^2? those are the definitions of the pythagorean triple correct? can anyone say yea or nay? if nay, how can i make it correct?

You have the general form of the triples correctly stated, but they are not of the form 3k, 4k, 5k, consider

[tex]12^2+5^2=13^2, or 20^2+ 21^2 =29^2.[/tex]

How could you argue about that? Is 29 a multiple of 5, is 13?
 
1+1=1 said:
the wording is correct.

No it is not. Write down exactly what you've been asked.
 
IF you have copied the problem exactly then the problem is wrong:

As has been pointed out, 5, 12, 13 is a Pythagorean triple (25+ 144= 169) but is NOT of the form 3k, 4k, 5k, even for k not an integer: if 3k= 5 then k= 5/3 but 4(5/3)= 20/3 is NOT equal to 12.

It MAY be that the problem asks you to show that any triple of the form (3k, 4k, 5k) IS a Pythagorean triple. That exactly the opposite of what you wrote and is very easy to show.
 
is the question demanding a counter-example here?
 

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