Help finding the error in the proof  TOPIC_SOLVED

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Help finding the error in the proof

Postby tonyc1970 on Mon Sep 02, 2013 4:24 pm

Hi,

First let me say that I am not looking for the answer, just a few hints to help me get started. I want to make sure I understand this as much as possible.

The problem states for me to identify the error in the proof:

Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = +- 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefor m = +- 1.

Thanks for any help,

Tony
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Re: Help finding the error in the proof

Postby nona.m.nona on Mon Sep 02, 2013 9:13 pm

tonyc1970 wrote:...identify the error in the proof:

Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = +- 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefor m = +- 1.

The proof makes little sense, as it appears to start off by stating what it then concludes. What statement is this proof meant to be proving?

Also, what have you used in your attempts to find errors? What progress have you made? Kindly please be complete. Thank you.
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Re: Help finding the error in the proof

Postby tonyc1970 on Mon Sep 02, 2013 10:14 pm

Hi nona.m.nona,

This is the proof exactly as it is written in my book,

Code: Select all
Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.


I am to find the error in this proof but I am not sure where to start.

Thanks
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Re: Help finding the error in the proof

Postby nona.m.nona on Mon Sep 02, 2013 11:46 pm

tonyc1970 wrote:This is the proof exactly as it is written in my book,

Code: Select all
Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.

You present the above as the proof. But of what is this the proof? What is the exact statement of the proposed theorem for which the above is the proof?
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Re: Help finding the error in the proof

Postby tonyc1970 on Mon Sep 02, 2013 11:53 pm

This is the exact statement, as I have already pointed out. I cannot provide any more information since this is all I have.

Code: Select all
Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1. If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.
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  TOPIC_SOLVED

Postby stapel_eliz on Tue Sep 03, 2013 10:12 am

tonyc1970 wrote:This is the exact statement, as I have already pointed out. I cannot provide any more information since this is all I have.

Is it possible that the thing you're trying to prove is the first bit? And then the rest is the attempted proof?

"Theorem": Let u, m, n be three integers. If u|mn and gcd(u,m) = 1, then m = (+ or -) 1.

"Proof": If gcd(u,m ) = 1, then 1 = us + mt for some integers s,t. If u|mn, then us = um for some integer s. Hence, 1 = mn + mt = m(s + t), which implies that m|1, and therefore m = (+ or -) 1.

The first bit (in blue above) is clearly not true. It makes more sense, I think, if one assumes the green part (above) is the "proof" part.

My suggestion would be to attack the second statement in the green part. :wink:
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