[seqfan] Re: Pi Day Question
Henry Gould
gould at math.wvu.edu
Sun Mar 15 00:32:22 CET 2009
I believe that the number
0.12345678910111213141516171819202122232425262727293-031.. . .
was proved to be normal many, many years ago. I heard a rumor fifty-two
years
ago via my number theory teacher, the late Alfred T. Brauer (at Univ. of
N.C.)
that perhaps a Russian mathematician had proved that the number
0.23571113171923293137 . . . formed by joxtaposing the digits of the prime
number sequence, is normal; however I do not recall finding a reference.
Can any Sequence Fanatic answer this and can anyone cite other specific
numbers
which are normal?
Henry Gould
= = = = = = =
Jonathan Post wrote:
> In mathematics, a normal number is a real number whose digits in every
> base show a uniform distribution, with all digits being equally
> likely, all pairs of digits equally likely, all triplets of digits
> equally likely, etc.
>
> While a general proof can be given that almost all numbers are normal,
> this proof is not constructive and only very few concrete numbers have
> been shown to be normal. It is for instance widely believed that the
> numbers √2, π, and e are normal, but a proof remains elusive.
>
> On Sat, Mar 14, 2009 at 1:07 PM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
>
>> [Whoops. I accidently sent this to seqfan at seqfan.com. Resending to correct address now...]
>>
>> Happy Pi Day, everybody! (At least in the US, where it is still March 14 and where the date is written 3/14.)
>>
>> A pi question:
>>
>> Consider the simple continued faction of pi. (The terms of which are sequence A001203.)
>>
>> The comment at the related sequence A032523 (A032523(n) = the index of the first occurrence of n in A001203) suggests that it is not known for certain that every positive integer occurs in the simple continued fraction of pi.
>>
>>
>> Can it be said, however, that either: it is known that each positive integer that does occur occurs infinitely often; OR it is known that at least some integers occur finitely often?
>>
>>
>> I am not even sure, from what little I have read, that if is known that 1 occurs infinitely often in the continued fraction of pi.
>>
>> Can anyone enlighten me (and us)?
>>
>> Thanks,
>> Leroy Quet
>>
>>
>>
>>
>>
>>
>>
>>
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>>
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