[seqfan] Re: Pi Day Question
Henry Gould
gould at math.wvu.edu
Sun Mar 15 00:34:34 CET 2009
Oops (my lousy typing) that is supposed to be
0.12345678910111213141516171819202122232425262728293031.. . .
Henry Gould wrote:
> 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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>>>
>>>
>>>
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>
>
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