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Does the golden ratio $\phi$ have infinitely many prime prefixes? That is, is the sequence A064117 infinite?

I'm guessing that $\phi$ does have infinitely many prime prefixes if $\phi$ is normal, similar to this comment on this question about $\pi$.

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  • $\begingroup$ I don't think there is much of a difference between the questions, really, given our limited state of knowledge. The same heuristic arguments could apply to any number where the digits "seem random" (and I believe $\phi$ is conjectured to be normal.) $\endgroup$ – Jair Taylor Jun 2 '17 at 0:46

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