Re: A fairly simple proposition

From: I.M.Davidson (sttscitrans_at_tesco.net)
Date: 11/14/04 

Date: 14 Nov 2004 12:56:44 -0800

dgoncz@aol.com ( Doug Goncz ) wrote in message news:<20041114082048.07117.00000599@mb-m10.aol.com>...
> Well, I recall something like if gcd(a,b,c)=1 then gcd(a+b, c-a, c-b)=1...
>
> A counterexample woudl certainly be useful, but I'd like to hear some theory.
> It's really pretty bare. Just the two conditions gcd(a,b,c)=1 and a<b<c<(a+b)
> do produce c-a !== 0 (mod b) OR c-b !== 0 in my searches to c=101. But why
> would that be? I haven't found a counterexample.
>
> What's the symbol for OR? V? or \/?
>
> Restating:
>
> Given
> 0<a,b,c
> a<b<c<(a+b)
> gcd(a,b,c)=1
>
> Proposition:
> c-a !== 0 (mod b) \/ c-b !== 0 (mod a)
>
> Proposition true? Counterexample? Discussion?

c-a == 0 (mod b) => c = a +mb
c-b == 0 (mod a) => c = b +na

a < b < c <a+b

a< b < a +mb < a+b
a< b < b +na < a+b
 => 0 < m,n < 1
so c cannot be an integer.

Relevant Pages

  • Re: A fairly simple proposition
    ... Well, I recall something like if gcd=1 then gcd(a+b, c-a, c-b)=1... ... A counterexample woudl certainly be useful, but I'd like to hear some theory. ... Proposition true? ...
    (sci.math)
  • Re: Brittish and US SF
    ... that I realized his argument was limited to denying any logic or counterexample and restating his assertion. ...
    (rec.arts.sf.written)