Re: Logarithmic Spirals

Robert Israel wrote:

thus sin (a pi x) b^x would not be algebraic.

But are there any such points?

One thing that can be said, anyway, is that _both coordinates_ are
algebraic exactly if x is rational, which is what I actually proved.

No, you didn't. You didn't say anything about what happens when x is
transcendental.

So what I have shown is

x rational -> both coordinates algebraic
x irrational alg. -> at least one coord. transcendental
x transcendental -> we know nothing

Is this right?

Andrew Usher

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