I did some algebraic transformations on equations with multiple real variables $x_i$, and I'd like to check whether the transformed equation is still valid. The equations are basically only rational functions with rational coefficients.
I found https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem and made a guess that using $x_i=\exp{\sqrt{p_i}}$ ($p_i$ is $i$-th prime number) for trial numbers would be a good idea, since plain arithmetic operations are unlikely to introduce spurious equalitities?! So, I plug in the trial numbers and see if the equations still hold (given enough calculator precision).
Is my interpretation correct? Am I guaranteed to detect transformation errors this way, if my equations consist of basic arithmetics $+,-,*,/$ only? Or under what circumstances would the test be reliable?
You cannot always guarantee the correctness by testing finitely many cases or even a countably infinite sequence of cases, as it may be that the error only occurs on some other values. It can be done in some instances. For example, for a broad class of continuous transformations it may be sufficient to check all rational values (based on the fact that if F and G are continuous real functions and F(x)=G(x) for all rational x , then F=G ).It depends on what you mean by a transformation.