
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
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<b, v<a such aubv = 2e 3) there exist two odd positive integers, x<b, y<a such byax = 4e For example 3 and 7 : 73 = 4 3.37 = 2 3 and 5 : 53 =2 3.35 = 4 5 and 7 : 75 = 2 5.5  7.3 = 4 11 and 15 : 15  11 = 4 11.7  15.5 = 2 7 and 17 : 7.3  17 = 4 17.3  7.7 = 2 Euzenius 
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 

Tags 
bezout, identity, odd, primes 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
primes and twin primes: Number between powers of 10  caters  Number Theory  67  March 19th, 2014 04:32 PM 
Primes in Z ?  gelatine1  Algebra  4  September 1st, 2013 10:09 PM 
n^2+1 n^2+n+1 primes  johnr  Number Theory  20  April 9th, 2013 05:49 PM 
primes  agustin975  Number Theory  11  March 10th, 2013 05:40 AM 
primes  Fernando  Number Theory  6  March 12th, 2012 08:36 AM 