Let $\sqrt 2 = 1.a_1a_2\dots _2$, and $\sqrt 3 = 1.b_1b_2\dots _2$. What can one say about the number $n = 0.c_1c_2\dots$ where $c_i = 1$ if $a_i = b_i$ and $0$ otherwise? There is no reason to believe that it is rational, but it isn't clear how to prove that it is irrational or transcendental.
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$\begingroup$ If n were rational, it might be possible to prove that the difference between the roots is also rational. $\endgroup$– The Masked AvengerCommented Jun 19, 2015 at 2:06
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$\begingroup$ I wonder if asking about $\sqrt{2}$ and $\sqrt{2}+\frac{1}3$ would be more tractable than $\sqrt{2}$ and $\sqrt{3}$ (I suspect that $\sqrt{2}$ being normal - or at least "random enough" - would at least tell us that that bitwise sum was irrational, though I lack proof) $\endgroup$– Milo BrandtCommented Jun 19, 2015 at 2:19
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$\begingroup$ So we ask about the "Nim sum" of $\sqrt{2}$ and $\sqrt{3}$: addition without carry in mod 2. $\endgroup$– Gerald EdgarCommented Jun 19, 2015 at 15:16
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