Author: fca
Date: 14:20:45 08/16/98
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On August 16, 1998 at 16:56:46, James B. Shearer wrote: >On August 16, 1998 at 16:04:44, fca wrote: > >>... pi is normal, which is >>why *I know* that somewhere in its decimal expansion are 58,124,760 consecutive >>"zero" digits, exactly. In fact, not just in one place, but in infinitely many >>such places. > > Is it in fact known that pi is normal? Reference please. It is inconceivable that it is not, but that is no proof. I'll try and dig a ref out for you, assuming its normality has now been established: I saw a fragment (the start) of a proof awaiting publication that did the rounds only a few years back, off-line. As you can imagine it was obscure so did not register. if your implication is right, it may well have bitten the dust since, for the usual reason. I'll use email unless you can find a way of re-chessing the thread ;-) "e" was shown to be normal c100 yrs back, in a well-validated proof, as you probably know, and 0.123456789101112131415161718192021...... is _far_ easier shown to be normal (though *not* entirely trivially IMO). Surely normal transcendentals are the "majority" of t's (if so, they are the majority of reals)? What proportion of (irrational - but this prefix is irrelevant to the answer) of non-t's are normal? <please ignore considerations of how hard it is to prove anything is normal, and use your instinct ;-) > btw, you might like to consider a really elegant proof that an irrational to an irrational power can be rational (hint start: SQRT(2) belongs to R-Q). Kind regards fca
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