use each digit (from the given base set) just once to produce the best possible approximation of Pi

[Christian Boyer]

Alexander, your problem reminds me of another problem:
	fractions using the same digits as their decimal representation.
Look at http://cboyer.club.fr/FractionsDigits.htm with for example the nice
	124983 / 576 = 216.984375

You will see that there is an open problem: who will provide at least one
solution, or a proof of impossibility, in base 4?

Christian.


-----Message d'origine-----
De : Alexander Povolotsky [mailto:apovolot at gmail.com] 
Envoyé : mardi 25 mars 2008 14:07
À : seqfan at ext.jussieu.fr
Objet : use each digit (from the given base set) just once to produce the
best possible approximation of Pi

Greetings to All,

I noticed that

689725314 / 219546387 = 76636146 / 24394043 = 3.141592642...

Of course the *quality of approximation" is very poor (16 digits "in"
are yielding just 8 accurate digits) but please note that in the ratio
below

6897253140 / 2195463870

all 10 digits of the decimal base (from 0 to 9) are used just once in
both numerator and denominator.
I hope that I found the *best* combination of digits for both
numerator and denominator (but I am notsure since I've done it late
night by hand).

I wonder if anyone would be interested to see what other (than decimal
) base systems could give with regards to the same approach of using
each digit (from the given base set) just once to produce the best
possible approximation of Pi.

If results are interesting, then one could compose two sequences (for
numerators and denominators) where a(10) will be correspondingly
6897253140 and 2195463870.

Regards
Alexander R. Povolotsky