Approximation of 2^M to 3^N /could this approach be useful?
From: Gottfried Helms (helms_at_uni-kassel.de)
Date: 09/27/04
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Date: Mon, 27 Sep 2004 00:30:02 +0000 (UTC)
Considering the approximation
a < | 2^M/3^N - 1 | (1)
where, with ß = log(3)/log(2)
M = ceil(ß *N) (2)
I found a representation of that problem in terms
of a hypergeometric series, which makes use of the
observation, that log(2) can be expressed in the
terms of the series for log(4/3).
Let
f = log(8/9) and (3)
g = log(2)
then I can reformulate the problem into a discussion
of differences of weigthed terms of f, something
like
with p_n = 2^M/3^N
log(p_n) = T_n * g/2 + N * f/2 (4)
where T_n = 2*M - 3*N
I get then, displayed in a Numbersystem base 3 and allowing rational
digits:
1 1 1 1 1 1
log(P_n) = T_n * 0. - 0 - 0 - 0 - 0 - 0 -- 0 .... (5)
1 3 5 7 9 11
1 1 1 1 1
- N * 0. 0 - 0 - 0 - 0 - 0 -- 0 ....
2 4 6 8 10
or combined as a hypergeometric series, replacing T_n by its composition
of M and N and equalling denominators,
(6)
12*M - 19*N 1 24*M - 39*N 1 36*M - 59*N 1
log(P_n)= ----------- *--- + ----------- *--- + ----------- *--- + ...
2 9^1 12 9^2 30 9^3
which is in a summation-notation:
N + 4*i*(3 M - 5 N) 1
log(P_n) = Sum ------------------- * --- (7)
i>0 2i * (2i - 1) 9^i
N 5 N - 3 M 1
log(P_n) = Sum ( ---------- - 2 ----------- ) * --- (8)
i>0 2i (2i-1) 2i - 1 9^i
To use that for the approximation-question, I need to find arguments,
how it could be shown, how far an amount of leading terms extinguish
each other to become a contiguous group of zeros from the left to
a certain maximal position i (depending on N).
>>From the 1-cycle-question of the collatz-problem, as well as
from an inequality steming from the waring-problem, I get as
a very likely, but yet unproven, statement
1/2^N < frac(2^M/3^N) (9)
which only requires that as first the about N/3'th term of such
a series need different from zero to have this inequality satisfied.
Some computations on my series indicate, that the first term, which
is too small to produce a carry to its leading terms, is already at
about the N/35'th term.
Now the condition for all leading terms down to the N/3'th to be
zero is different from that to not be able to produce a carry;
but I think, it could be an interesting approach to get a view
into the matter, since a contiguous group of such terms can only
be zero by getting extinguished by carries from other terms
(always with given N and M).
Could someone argue for/against the value of this consideration,
or indicate a resource, if this representation was discussed
elsewhere?
Regards -
Gottfried Helms
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