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

[Artur]

Dear Alexander,
Your results are related to
http://www.research.att.com/~njas/sequences/A138343

If you want improove approximation by rational fraction p/q of number Pi 
exactly to n digits
what meant that 10^n<Pi-p/q<10^n+1
you have to defined yet which function do you  minimalize. These 
functions can be:
min(p+q),min(pq),min(p^2+q^2) etc.
also you can looking for only lower limits    
p/q<Pi
or upper limits
p/q>Pi
The best approximations occured for truncated reversed continued fractions
Mathematica code:
b = {}; Do[c = FromContinuedFraction[ContinuedFraction[1/Pi, n]];
AppendTo[b, c], {n, 1, 20}]; b
but in this case lower limits occured for odd n and upper for even n
for lower limits
b = {}; Do[c = FromContinuedFraction[ContinuedFraction[1/Pi, n]];
AppendTo[b, c], {n, 1, 20,2}]; b
for upper limits
b = {}; Do[c = FromContinuedFraction[ContinuedFraction[1/Pi, n]];
AppendTo[b, c], {n, 2, 20,2}]; b

If you define minimalzation function and variant of limit generally 
happend cases that better approximation will occured for smaller 
minimalization function which one approximated Pi to e.g. 10 digits that 
8 (with forced condition "exactly to n digits")
e.g. for Pi in decimal system 355/113 approximated to 6 places after 
comma and if you forced exactly to 5 digital places you need uses bigger 
p and q as to 6 places.

But very suprising for me was that for decimal system existed numbers 
(1+Sqrt[5])/2 and Sqrt[2] where occured increasing
set of fractions (numerators and denominators only decreased) where we 
improoved in next step agreement with rational number improoved about 1 
next digit (I say that decimale system is accurate for (1+Sqrt[5])/2 and 
Sqrt[2]))
And very difficult question. Existem positional system such that we will 
obtained successive approximation  for Pi ?
These is very difficult matehamtical problem connected with reversed JIS 
problem. JIS function reversed number in  particular counting system  on 
set of integers,  reversed JIS need  finding for given decreasing set of 
integers number and base of positional system. I'm working on this 
problem since many years with pure results.

BEST WISHES
ARTUR











Alexander Povolotsky pisze:
> 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
>
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