[FOM] Primitive recursive reals
Timothy Y. Chow
tchow at alum.mit.edu
Thu Apr 20 18:02:51 EDT 2006
Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> QUESTION: Is e + pi primitive recursive in base 2, base 10, or even (as
> expected) primitive recursive in every base?
This looks difficult. It isn't known whether e+pi is irrational. Of
course, one might still be able to answer the above question affirmatively
along the lines of, "If e+pi is rational then it is primitive recursive in
every base; if e+pi is irrational then...." However, given that we don't
know how to control the diophantine approximation properties of e+pi well
enough to prove its irrationality, the bare assumption that e+pi is in
fact irrational is unlikely to give enough of an extra boost to get the
primitive recursive bound that it seems one needs to answer Harvey
Friedman's question.
Tim
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