I was reading about linear Diophantine equation of the form:
ax+by+cz=0
The equation has infinitely many integer solutions x,y,zϵZ as long as the coefficients a,b,c are integer numbers. Could someone refer me to an article, book or other source where the equation is treated without the assumption that the coefficients are integers?
Thanks for any helpful references.
If the coefficients are rationally related to each other you can multiply by a constant to make them all integers. For example you can go from $$\sqrt 2 x + \frac 32\sqrt 2 y + \sqrt 8 z=0$$ to $$2x+3y+4z=0$$ so nothing is lost in assuming the coefficients are integers.
If they are not rationally related you need a very special relationship between them. For example $$x+\sqrt 2 y +(\sqrt 2-1)z=0$$ does have integer solutions, but you cannot clear the irrationals by multiplication/division. We could come up with cases that were not so obvious. I am not aware of study of these cases because we then are solving in the reals, not the integers.