Re: Whether solution exists
- From: Ross <rmillika@xxxxxxxxxxx>
- Date: Tue, 30 Dec 2008 10:12:48 -0800 (PST)
On Dec 30, 4:57 am, nyg...@xxxxxxxxx wrote:
hi,
i'm finding it tough to prove why a solution always exists. I wrote
some code and found that it always exists for random permutations.
Can you find non negative integers 0<=a1,a2,a3,a4,a5,a6 <100 such
that
{ sum (i=0 to 6) |xi| } <> 0
and
| x0 + { sum(i=1 to 6) xi*sqrt(ai) } | < 10^-4 doesn't have solution
for integers |xi| < 10 .
I tried writing some code but i always find a solution. Can anyone
tell me why i can always find a solution in this case or do we have a
integer vector ai so that the solution doesn't exist.
Below is my code . It can be useful if you want to check try couple of
values.http://codepad.org/NCljunTJ
I doubt you will find a proof except by exhaustive testing. Clearly
you can
find a number to replace 10^-4 which will make it fail.
It is not surprising that there is usually an answer. If you consider
the
fractional part of xi*sqrt(ai), they will be scattered around (0,1)
reasonably
uniformly. You have 18 choices for each xi, so there are 18^6
fractional
parts for each set of ai's. This is about 34 million tries and 1/5000
of
the tries should succeed-if I read you correctly there is no bound on
x0,
so if the fractional part is in (0,10-4) or in (.9999,1) you succeed.
For a random set of xi's you have a .9998^(18^6) chance of failure,
which
is about 10^-29572.
If any ai is a square or two are equal there will be a trivial
solution, so
you have C(90,6) sets of ai, or about 623 million. The chance of
failure
even with this many is vanishingly small.
Use M for million below.
Ignoring small factors like e, we can estimate how small we can make
your 10^-4
and still be likely to always have a solution. Let b be the allowable
miss
distance (instead of 10^-4). Then we want (1-2b)^(18^6)*623M=.5.
Taking natural
logs of both sides, ignoring ln(2), and using ln(1-x)=-x for small x,
we get
b=ln(623M)/68M=20/68M so we would expect always to have a solution
withing 10^-6
or so.
.
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