I would like to examine whether the following claim is true:
For the following equation:
ax+by=gcd(a,b)
where a,b,∈N and x,y∈Z, there exist x1,y1∈Z such that x1≠x,y1≠y and
ax1+by1=gcd(a,b)
I believe it holds and that it is also possible to express analytically x1,y1 in terms of x,y. I have reached the following point:
ax+by=gcd(a,b)⇔ax+ax1−ax1+by+by1−by1=gcd(a,b)⇔a(x−x1)+b(y−y1)+(ax1+by1)=gcd(a,b)⇔a(x−x1)+b(y−y1)=0
where I used the fact that (ax1+by1)=gcd(a,b). After distinguishing cases I end up with $x>x_1, y
There must be some wrong step in my proof but I cannot figure it out.
No comments:
Post a Comment