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 July 11th, 2009, 02:59 PM #1 Member   Joined: Jul 2009 Posts: 55 Thanks: 0 Bezout's identity and odd co primes Good evening, Let a and b two odd positive integers >1 and e = +1 or -1, is it true that... ...these propositions are equivalent : 1) a and b are co primes 2) there exist two odd positive integers, u
 July 12th, 2009, 04:02 AM #2 Member   Joined: Jul 2009 Posts: 55 Thanks: 0 Re: Bezout's identity and odd co primes And what about : 2') there exists one and only one pair (u,v)... 3') ... one and only one pair (x,y)... instead of (2) and (3) Euzenius
 July 13th, 2009, 07:39 AM #3 Member   Joined: Jul 2009 Posts: 55 Thanks: 0 Re: Bezout's identity and odd co primes But, 9x3 - 5x5 = 2 9 - 5 = 4 No crossing, I was surely tired... Euzenius
 July 16th, 2009, 10:41 AM #4 Member   Joined: Jul 2009 Posts: 55 Thanks: 0 Re: Bezout's identity and odd co primes Hello, But in fact if p,q are odd positive integers >1, these propositions are equivalent : 1) p and q are co primes 2) there exists only one pair ( s, t), q>s>0, p>t>0 such sp - tq = 1 3) there exist only one pair (u, v), q>u>0, p>v>0 such vq - up = 1 As you can see s+u = q and t+v = p (but the linear system 4x4 in s,t,u,v is linked, determinant of the matrix is zero) But could you prove that sv - tu = +1 or -1 ? (Or find a counter example ?) Sorry I don't get any proof... Euzenius

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