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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?

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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.

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  • $\begingroup$ But my function class seems pretty trivial. It's mostly rational functions. And I only do basic high school rearrangements. Do this help to prove anything? $\endgroup$
    – Gerenuk
    Aug 22 '15 at 19:30
  • $\begingroup$ If p,q,r,s are polynomials then p/q=r/s implies pr=qs. If the degree deg(pr)=deg(qs) =n, you can see whether pq and rs agree at n+1 points because if two polynomials of degree n agree at n+1 points they are equal everywhere.. $\endgroup$ Aug 26 '15 at 18:36
  • $\begingroup$ The question is still whether a single test is enough. Can you give an example equation which includes $\exp\sqrt{2}$ and $\exp\sqrt{3}$, has only basic arithmetics and has conflicting truth values after transformations? $\endgroup$
    – Gerenuk
    Aug 27 '15 at 5:23

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