| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 67.1 | | HARE::STAN | | Mon May 14 1984 19:59 | 4 |
| I've tried changing 2 to 3 and looked at the result in base 3.
I have some interesting findings but nothing definitive yet
on a generalization.
|
| 67.2 | | LAMBDA::VOSBURY | | Wed Jul 18 1984 13:53 | 71 |
| In what follows [ x ] is the floor function or the integer part of the real
number x and { x } is the fractional part, i.e. x = [ x ] + { x }.
Given the recursion:
x(1) = 1, x(k+1) = [ sqrt(b) * ( x(k) + c ) ]
consider:
n+1 n n+1 n n
x(k) = [ b + sqrt(b) * b ] = b + sqrt(b) * b - { sqrt(b) * b }
hence:
n+1 n+1 n
x(k+1) = [ sqrt(b) * b + b - sqrt(b) * { sqrt(b) * b } + sqrt(b) * c ]
n+1 n+1
x(k+1) = [ sqrt(b) * b + b ] +
n+1 n
[ { sqrt(b) * b } - sqrt(b) * { sqrt(b) * b } + sqrt(b) * c ]
Now let's look at the possibility of making the second term vanish, i.e.
making:
n+1 n
0 <= { sqrt(b) * b } - sqrt(b) * { sqrt(b) * b } + sqrt(b) * c < 1
now:
n n n
b * { sqrt(b) * b } = [ b * { sqrt(b) * b } ] + { b * { sqrt(b) * b } }
n n+1
= [ b * { sqrt(b) * b } ] + { sqrt(b) * b }
= i + y
where 0 <= i <= (b-1) and 0 < y < 1. Hence:
0 <= y - ( i + y ) / sqrt(b) + sqrt(b) * c < 1
0 <= sqrt(b) * y - i - y + b * c < sqrt(b)
i - y * ( sqrt(b) - 1 ) <= b * c < sqrt(b) + i - y * ( sqrt(b) - 1 )
Assuming sqrt(b) > 1 we find b - 1 <= b * c <= 1 which can only be
satisfied for b = 2 and c = 1/2. Thus if
n+1 n
x(k+1) = [ sqrt(2) * ( x(k) + 1/2 ) ] and x(k) = [ 2 + sqrt(2) * 2 ]
n+1 n+1
then x(k+1) = [ sqrt(2) * 2 + 2 ]. It follows from a similar, slightly
n+1 n+1
simpler argument, that if x(k+1) = [ sqrt(2) * 2 + 2 ] then
n+2 n+1
x(k+2) = [ 2 + sqrt(2) * 2 ]. This plus the fact that
0 0-1
x(1) = 1 = [ 2 + sqrt(2) * 2 ] forms the basis for an inductive proof
of the original proposition. The argument also shows that a recursion for
sqrt(b), b>2 must be more complex than x(n+1) = [ sqrt(b) * ( x(n) + c ) ].
Mike.
|
| 67.3 | | TURTLE::GILBERT | | Thu Jul 19 1984 01:02 | 20 |
| I'm impressed and enlightened by Mike's response.
It states a generalization of the original recurrence, proposes a solution to
the recurrence (the relationship between k and n less than obvious) that is
consistent with producing the square root (d(n) in the original note), proves
that this produces the square root only for the originally stated recurrence,
and concludes that a different generalization is required to similarly produce
other square roots.
I'm still having some trouble understanding parts of the previous response.
I don't understand the steps between the following statements:
i - y * ( sqrt(b) - 1 ) <= b * c < sqrt(b) + i - y * ( sqrt(b) - 1 )
Assuming sqrt(b) > 1 we find b - 1 <= b * c <= 1 which can only be
satisfied for b = 2 and c = 1/2.
A little more light, Mike?
- Gilbert
|
| 67.4 | | CACHE::VOSBURY | | Thu Jul 19 1984 15:06 | 38 |
| I was just rushing to a conclusion I originally came to by a tedious
examination of cases. In the form I gave in the response, I think I can
shortcut it some. Given:
i - y * ( sqrt(b) - 1 ) <= b * c < sqrt(b) + i - y * ( sqrt(b) - 1 )
and b > 1, 0 <= i <= (b-1), 0 < y < 1. (Another way to express the last
inequality is 0 + e <= y <= 1 - e, e > 0.) We're trying to find the
range of allowable values for c as a function of b. If we pick values of
i and y which MAXIMIZE the left hand expression (maximum i, minimum y) we have:
( b - 1 ) - e * ( sqrt( b ) - 1 ) < b - 1
so if we constrain c to satisfy b - 1 <= b * c we will satisfy the
left hand inequality in our original expression for all allowable values of
i and y. Likewise, if we pick (different) values of i and y to MINIMIZE the
right hand expression (minimum i, maximum y) we have:
sqrt(b) + 0 - ( 1 - e ) * ( sqrt(b) - 1 ) = 1 + e * ( sqrt(b) - 1 )
so if we constrain c to satisfy b * c <= 1 we will satisfy the
right hand inequality in our original expression for all allowable values of
i and y. Combining we get b - 1 <= b * c <= 1 which for integer
b > 1 can only be satisfied by b = 2 and c = 1/2.
As to how I discovered this, I stumbled across this note when I was thinking
about Stan's PI generating algorithm and I was struck by the similarity
between this and the Stan's method of generating the digits of PI by
successively multiplying PI by the powers of ten. In thinking about this
recursion I realized that if x(2k+1) - 2 * (2k-1) yields a bit of
sqrt(2), then x(2k+1) must contain ALL of the bits of PI produced to date
(plus an additional 1 stuck on the front.) Sure enough, when I wrote a
little program to calculate a few values of the sequence and print them
out in binary the pattern to the values became apparent. The next step
was to prove it and while I was at it, with little additional effort, I
could examine some additional cases too.
Mike.
|
| 67.5 | | TURTLE::GILBERT | | Thu Jul 19 1984 16:47 | 3 |
| Ahh. Then there may be other values of b and c that satisfy the inequality,
even with b an integer! (i.e., there may be values for which the true extrema
of those expressions are not even close to the extrema in the analysis).
|
| 67.6 | | LAMBDA::VOSBURY | | Thu Jul 19 1984 19:23 | 20 |
| Well, maybe but remember what those things are: i is just a radix-b "digit"
in the radix-b expansion of the sqrt(b) and y is just the fractional
part of sqrt(b) * b ** n. If you believe that the radix-b expansion of
sqrt(b) behaves like an infinite set of throws of a b-sided die, than you
would expect somewhere in that sequence to find the sub-sequence of "digits"
..., (b-1), 0, 0, 0, 0, 0, 0, 0, 0, ....
and so on for as many zeros as you like. This will drive the left hand
expression as close as never-mind to (b-1). Likewise, you would also
expect to find somewhere in there:
..., 0, (b-1), (b-1), (b-1), (b-1), (b-1), ...
which will drive the right hand expression as close to 1 as you like.
Of course, sqrt(b) is not a random number and it may be possible to prove that
those sub-sequences cannot exist but I wouldn't know how to go about it.
Mike.
|
| 67.7 | | LAMBDA::VOSBURY | | Fri Jul 27 1984 12:54 | 27 |
| Let b be an integer > 1 and R be any real number such that 0 <= R < b.
[ x ] is the floor function (integer part of x). Let:
a = 1
0
a = [ ( b + R ) * b ** [ log (a ) ] ]
n+1 b n
n n-1
Assume a = [ b + R * b ]. Then [ log (a ) ] = n and
n b n
n+1 n
a = [ ( b + R ) * b ** n ] = [ b + R * b ].
n+1
Now express a in base-b and strip off the leading "1" or equivalently
n+1
calculate a - b * a .
n+1 n
Mike.
|
| 67.8 | | TURTLE::GILBERT | | Mon Jul 30 1984 04:41 | 12 |
| Let x = 1, and let x = [ a [ b * x + b/2 ] + a/2 ], where b = 2/a.
0 n+1 n
inf -n
Let d = x - 2x , and let z = Sum d 2 .
n n-1 n n=0 n
The original problem gives z = sqrt(2) if a = sqrt(2).
Note also that z = sqrt(3)-1 if a = sqrt(3).
- Gilbert
|
| 67.9 | | HARE::STAN | | Wed Aug 01 1984 03:29 | 387 |
| We can rewrite Peter's amazing observation as follows:
Let [y] denote the floor of y.
Let <x,a> denote [ ax+ a/2 ] .
Now define the sequence x(n) recursively as
x(0) = 1
/ <x(n),a> if n is even
x(n+1) = <
\ <x(n),b> if n is odd
where b = 2/a .
Then we observe (by computer program), that if a is between 1 and 2 inclusive,
then the quantities
d(n) = x(n) - 2 x(n-2) ( n>1 )
are always equal to either 0 or 1. Thus we can use these digits to form
a binary representation of some real number. Actually, we form two real
numbers, from the even digits and the odd digits. Specifically,
----- ------
\ -[(n-1)/2] \ -[(n-1)/2]
Xodd = > d(n) 2 Xeven = > d(n) 2 .
/ /
----- ------
odd n even n
For some reason that I don't understand, these d's do not come out to be
just 0 and 1 if a is less than 1 or greater than 2.
I wrote a program that tabulated Xodd and Xeven as I varied N from 1 through 4
in steps of 0.01, setting a=sqrt(N) and b=2/a. See appendix for data.
By analyzing this data, I arrive at the conjecture that
for 1.0 < a < 1.5, Xodd = (2a-2)/(2-a) and Xeven = {Xodd}
for 1.5 < a < 2.0 Xodd = (4-2a)/(a-1) and Xeven = Xodd / 2
where {y} denotes the fractional part of y.
There are discontinuities at 1.0, 1.5, and 2.0.
Boy is this wierd and random!
If N=2, then a=sqrt(2), b=sqrt(2), and Xodd = sqrt(2) as in the original note.
If N=3, then a=sqrt(3), b=2/sqrt(3), and Xodd=sqrt(3)-1 as Peter noted.
Appendix I (data)
N= 1.000 A= 1.000000 B= 2.000000 XEVEN= 1.000000 XODD= 2.000000
N= 1.010 A= 1.004988 B= 1.990074 XEVEN= 0.010025 XODD= 0.010025
N= 1.020 A= 1.009950 B= 1.980295 XEVEN= 0.020101 XODD= 0.020101
N= 1.030 A= 1.014889 B= 1.970659 XEVEN= 0.030228 XODD= 0.030228
N= 1.040 A= 1.019804 B= 1.961161 XEVEN= 0.040408 XODD= 0.040408
N= 1.050 A= 1.024695 B= 1.951800 XEVEN= 0.050641 XODD= 0.050641
N= 1.060 A= 1.029563 B= 1.942572 XEVEN= 0.060927 XODD= 0.060927
N= 1.070 A= 1.034408 B= 1.933473 XEVEN= 0.071268 XODD= 0.071268
N= 1.080 A= 1.039230 B= 1.924501 XEVEN= 0.081665 XODD= 0.081665
N= 1.090 A= 1.044031 B= 1.915653 XEVEN= 0.092117 XODD= 0.092117
N= 1.100 A= 1.048809 B= 1.906925 XEVEN= 0.102627 XODD= 0.102627
N= 1.110 A= 1.053565 B= 1.898316 XEVEN= 0.113194 XODD= 0.113194
N= 1.120 A= 1.058301 B= 1.889822 XEVEN= 0.123820 XODD= 0.123820
N= 1.130 A= 1.063015 B= 1.881442 XEVEN= 0.134505 XODD= 0.134505
N= 1.140 A= 1.067708 B= 1.873172 XEVEN= 0.145250 XODD= 0.145250
N= 1.150 A= 1.072381 B= 1.865010 XEVEN= 0.156056 XODD= 0.156056
N= 1.160 A= 1.077033 B= 1.856953 XEVEN= 0.166925 XODD= 0.166925
N= 1.170 A= 1.081665 B= 1.849001 XEVEN= 0.177855 XODD= 0.177855
N= 1.180 A= 1.086278 B= 1.841149 XEVEN= 0.188850 XODD= 0.188850
N= 1.190 A= 1.090871 B= 1.833397 XEVEN= 0.199908 XODD= 0.199908
N= 1.200 A= 1.095445 B= 1.825742 XEVEN= 0.211032 XODD= 0.211032
N= 1.210 A= 1.100000 B= 1.818182 XEVEN= 0.222222 XODD= 0.222222
N= 1.220 A= 1.104536 B= 1.810715 XEVEN= 0.233479 XODD= 0.233479
N= 1.230 A= 1.109054 B= 1.803339 XEVEN= 0.244804 XODD= 0.244804
N= 1.240 A= 1.113553 B= 1.796053 XEVEN= 0.256198 XODD= 0.256198
N= 1.250 A= 1.118034 B= 1.788854 XEVEN= 0.267661 XODD= 0.267661
N= 1.260 A= 1.122497 B= 1.781742 XEVEN= 0.279195 XODD= 0.279195
N= 1.270 A= 1.126943 B= 1.774713 XEVEN= 0.290801 XODD= 0.290801
N= 1.280 A= 1.131371 B= 1.767767 XEVEN= 0.302479 XODD= 0.302479
N= 1.290 A= 1.135782 B= 1.760902 XEVEN= 0.314230 XODD= 0.314230
N= 1.300 A= 1.140175 B= 1.754116 XEVEN= 0.326056 XODD= 0.326056
N= 1.310 A= 1.144552 B= 1.747408 XEVEN= 0.337957 XODD= 0.337957
N= 1.320 A= 1.148913 B= 1.740777 XEVEN= 0.349935 XODD= 0.349935
N= 1.330 A= 1.153256 B= 1.734220 XEVEN= 0.361990 XODD= 0.361990
N= 1.340 A= 1.157584 B= 1.727737 XEVEN= 0.374123 XODD= 0.374123
N= 1.350 A= 1.161895 B= 1.721326 XEVEN= 0.386336 XODD= 0.386336
N= 1.360 A= 1.166190 B= 1.714986 XEVEN= 0.398629 XODD= 0.398629
N= 1.370 A= 1.170470 B= 1.708715 XEVEN= 0.411004 XODD= 0.411004
N= 1.380 A= 1.174734 B= 1.702513 XEVEN= 0.423461 XODD= 0.423461
N= 1.390 A= 1.178983 B= 1.696378 XEVEN= 0.436002 XODD= 0.436002
N= 1.400 A= 1.183216 B= 1.690309 XEVEN= 0.448628 XODD= 0.448628
N= 1.410 A= 1.187434 B= 1.684304 XEVEN= 0.461339 XODD= 0.461339
N= 1.420 A= 1.191638 B= 1.678363 XEVEN= 0.474138 XODD= 0.474138
N= 1.430 A= 1.195826 B= 1.672484 XEVEN= 0.487024 XODD= 0.487024
N= 1.440 A= 1.200000 B= 1.666667 XEVEN= 0.500000 XODD= 0.500000
N= 1.450 A= 1.204159 B= 1.660910 XEVEN= 0.513066 XODD= 0.513066
N= 1.460 A= 1.208305 B= 1.655212 XEVEN= 0.526224 XODD= 0.526224
N= 1.470 A= 1.212436 B= 1.649572 XEVEN= 0.539475 XODD= 0.539475
N= 1.480 A= 1.216553 B= 1.643990 XEVEN= 0.552819 XODD= 0.552819
N= 1.490 A= 1.220656 B= 1.638464 XEVEN= 0.566259 XODD= 0.566259
N= 1.500 A= 1.224745 B= 1.632993 XEVEN= 0.579796 XODD= 0.579796
N= 1.510 A= 1.228821 B= 1.627577 XEVEN= 0.593430 XODD= 0.593430
N= 1.520 A= 1.232883 B= 1.622214 XEVEN= 0.607164 XODD= 0.607164
N= 1.530 A= 1.236932 B= 1.616904 XEVEN= 0.620997 XODD= 0.620997
N= 1.540 A= 1.240967 B= 1.611646 XEVEN= 0.634933 XODD= 0.634933
N= 1.550 A= 1.244990 B= 1.606439 XEVEN= 0.648971 XODD= 0.648971
N= 1.560 A= 1.249000 B= 1.601282 XEVEN= 0.663114 XODD= 0.663114
N= 1.570 A= 1.252996 B= 1.596174 XEVEN= 0.677363 XODD= 0.677363
N= 1.580 A= 1.256981 B= 1.591115 XEVEN= 0.691719 XODD= 0.691719
N= 1.590 A= 1.260952 B= 1.586103 XEVEN= 0.706184 XODD= 0.706184
N= 1.600 A= 1.264911 B= 1.581139 XEVEN= 0.720759 XODD= 0.720759
N= 1.610 A= 1.268858 B= 1.576221 XEVEN= 0.735446 XODD= 0.735446
N= 1.620 A= 1.272792 B= 1.571348 XEVEN= 0.750246 XODD= 0.750246
N= 1.630 A= 1.276715 B= 1.566521 XEVEN= 0.765160 XODD= 0.765160
N= 1.640 A= 1.280625 B= 1.561738 XEVEN= 0.780191 XODD= 0.780191
N= 1.650 A= 1.284523 B= 1.556998 XEVEN= 0.795339 XODD= 0.795339
N= 1.660 A= 1.288410 B= 1.552301 XEVEN= 0.810607 XODD= 0.810607
N= 1.670 A= 1.292285 B= 1.547646 XEVEN= 0.825996 XODD= 0.825996
N= 1.680 A= 1.296148 B= 1.543033 XEVEN= 0.841507 XODD= 0.841507
N= 1.690 A= 1.300000 B= 1.538462 XEVEN= 0.857143 XODD= 0.857143
N= 1.700 A= 1.303840 B= 1.533930 XEVEN= 0.872905 XODD= 0.872905
N= 1.710 A= 1.307670 B= 1.529438 XEVEN= 0.888794 XODD= 0.888794
N= 1.720 A= 1.311488 B= 1.524986 XEVEN= 0.904814 XODD= 0.904814
N= 1.730 A= 1.315295 B= 1.520572 XEVEN= 0.920964 XODD= 0.920964
N= 1.740 A= 1.319091 B= 1.516196 XEVEN= 0.937248 XODD= 0.937248
N= 1.750 A= 1.322876 B= 1.511858 XEVEN= 0.953667 XODD= 0.953667
N= 1.760 A= 1.326650 B= 1.507557 XEVEN= 0.970223 XODD= 0.970223
N= 1.770 A= 1.330413 B= 1.503292 XEVEN= 0.986918 XODD= 0.986918
N= 1.780 A= 1.334166 B= 1.499063 XEVEN= 0.003753 XODD= 1.003753
N= 1.790 A= 1.337909 B= 1.494870 XEVEN= 0.020732 XODD= 1.020732
N= 1.800 A= 1.341641 B= 1.490712 XEVEN= 0.037855 XODD= 1.037855
N= 1.810 A= 1.345362 B= 1.486588 XEVEN= 0.055125 XODD= 1.055125
N= 1.820 A= 1.349074 B= 1.482499 XEVEN= 0.072545 XODD= 1.072545
N= 1.830 A= 1.352775 B= 1.478443 XEVEN= 0.090115 XODD= 1.090115
N= 1.840 A= 1.356466 B= 1.474420 XEVEN= 0.107839 XODD= 1.107839
N= 1.850 A= 1.360147 B= 1.470429 XEVEN= 0.125718 XODD= 1.125718
N= 1.860 A= 1.363818 B= 1.466471 XEVEN= 0.143755 XODD= 1.143755
N= 1.870 A= 1.367479 B= 1.462545 XEVEN= 0.161952 XODD= 1.161952
N= 1.880 A= 1.371131 B= 1.458650 XEVEN= 0.180312 XODD= 1.180312
N= 1.890 A= 1.374773 B= 1.454786 XEVEN= 0.198837 XODD= 1.198837
N= 1.900 A= 1.378405 B= 1.450953 XEVEN= 0.217528 XODD= 1.217528
N= 1.910 A= 1.382027 B= 1.447149 XEVEN= 0.236390 XODD= 1.236390
N= 1.920 A= 1.385641 B= 1.443376 XEVEN= 0.255424 XODD= 1.255424
N= 1.930 A= 1.389244 B= 1.439632 XEVEN= 0.274632 XODD= 1.274632
N= 1.940 A= 1.392839 B= 1.435916 XEVEN= 0.294018 XODD= 1.294018
N= 1.950 A= 1.396424 B= 1.432230 XEVEN= 0.313584 XODD= 1.313584
N= 1.960 A= 1.400000 B= 1.428571 XEVEN= 0.333333 XODD= 1.333333
N= 1.970 A= 1.403567 B= 1.424941 XEVEN= 0.353268 XODD= 1.353268
N= 1.980 A= 1.407125 B= 1.421338 XEVEN= 0.373391 XODD= 1.373391
N= 1.990 A= 1.410674 B= 1.417762 XEVEN= 0.393705 XODD= 1.393705
N= 2.000 A= 1.414214 B= 1.414214 XEVEN= 0.414214 XODD= 1.414214
N= 2.010 A= 1.417745 B= 1.410691 XEVEN= 0.434919 XODD= 1.434919
N= 2.020 A= 1.421267 B= 1.407195 XEVEN= 0.455825 XODD= 1.455825
N= 2.030 A= 1.424781 B= 1.403725 XEVEN= 0.476935 XODD= 1.476935
N= 2.040 A= 1.428286 B= 1.400280 XEVEN= 0.498251 XODD= 1.498251
N= 2.050 A= 1.431782 B= 1.396861 XEVEN= 0.519776 XODD= 1.519776
N= 2.060 A= 1.435270 B= 1.393466 XEVEN= 0.541515 XODD= 1.541515
N= 2.070 A= 1.438749 B= 1.390096 XEVEN= 0.563471 XODD= 1.563471
N= 2.080 A= 1.442221 B= 1.386750 XEVEN= 0.585646 XODD= 1.585646
N= 2.090 A= 1.445683 B= 1.383429 XEVEN= 0.608045 XODD= 1.608045
N= 2.100 A= 1.449138 B= 1.380131 XEVEN= 0.630671 XODD= 1.630671
N= 2.110 A= 1.452584 B= 1.376857 XEVEN= 0.653528 XODD= 1.653528
N= 2.120 A= 1.456022 B= 1.373606 XEVEN= 0.676619 XODD= 1.676619
N= 2.130 A= 1.459452 B= 1.370377 XEVEN= 0.699949 XODD= 1.699949
N= 2.140 A= 1.462874 B= 1.367172 XEVEN= 0.723520 XODD= 1.723520
N= 2.150 A= 1.466288 B= 1.363989 XEVEN= 0.747338 XODD= 1.747338
N= 2.160 A= 1.469694 B= 1.360828 XEVEN= 0.771406 XODD= 1.771406
N= 2.170 A= 1.473092 B= 1.357688 XEVEN= 0.795729 XODD= 1.795729
N= 2.180 A= 1.476482 B= 1.354571 XEVEN= 0.820310 XODD= 1.820310
N= 2.190 A= 1.479865 B= 1.351475 XEVEN= 0.845155 XODD= 1.845155
N= 2.200 A= 1.483240 B= 1.348400 XEVEN= 0.870266 XODD= 1.870266
N= 2.210 A= 1.486607 B= 1.345346 XEVEN= 0.895650 XODD= 1.895650
N= 2.220 A= 1.489966 B= 1.342312 XEVEN= 0.921311 XODD= 1.921311
N= 2.230 A= 1.493318 B= 1.339299 XEVEN= 0.947252 XODD= 1.947252
N= 2.240 A= 1.496663 B= 1.336306 XEVEN= 0.973481 XODD= 1.973481
N= 2.250 A= 1.500000 B= 1.333333 XEVEN= 1.000000 XODD= 2.000000
N= 2.260 A= 1.503330 B= 1.330380 XEVEN= 0.986769 XODD= 1.973539
N= 2.270 A= 1.506652 B= 1.327447 XEVEN= 0.973742 XODD= 1.947483
N= 2.280 A= 1.509967 B= 1.324532 XEVEN= 0.960912 XODD= 1.921823
N= 2.290 A= 1.513275 B= 1.321637 XEVEN= 0.948275 XODD= 1.896550
N= 2.300 A= 1.516575 B= 1.318761 XEVEN= 0.935827 XODD= 1.871654
N= 2.310 A= 1.519868 B= 1.315903 XEVEN= 0.923564 XODD= 1.847127
N= 2.320 A= 1.523155 B= 1.313064 XEVEN= 0.911481 XODD= 1.822962
N= 2.330 A= 1.526434 B= 1.310244 XEVEN= 0.899574 XODD= 1.799148
N= 2.340 A= 1.529706 B= 1.307441 XEVEN= 0.887840 XODD= 1.775680
N= 2.350 A= 1.532971 B= 1.304656 XEVEN= 0.876275 XODD= 1.752550
N= 2.360 A= 1.536229 B= 1.301889 XEVEN= 0.864874 XODD= 1.729749
N= 2.370 A= 1.539480 B= 1.299140 XEVEN= 0.853635 XODD= 1.707271
N= 2.380 A= 1.542725 B= 1.296407 XEVEN= 0.842554 XODD= 1.685108
N= 2.390 A= 1.545962 B= 1.293692 XEVEN= 0.831628 XODD= 1.663255
N= 2.400 A= 1.549193 B= 1.290994 XEVEN= 0.820852 XODD= 1.641705
N= 2.410 A= 1.552417 B= 1.288313 XEVEN= 0.810225 XODD= 1.620450
N= 2.420 A= 1.555635 B= 1.285649 XEVEN= 0.799743 XODD= 1.599486
N= 2.430 A= 1.558846 B= 1.283001 XEVEN= 0.789403 XODD= 1.578805
N= 2.440 A= 1.562050 B= 1.280369 XEVEN= 0.779201 XODD= 1.558403
N= 2.450 A= 1.565248 B= 1.277753 XEVEN= 0.769136 XODD= 1.538273
N= 2.460 A= 1.568439 B= 1.275153 XEVEN= 0.759205 XODD= 1.518409
N= 2.470 A= 1.571623 B= 1.272570 XEVEN= 0.749404 XODD= 1.498807
N= 2.480 A= 1.574802 B= 1.270001 XEVEN= 0.739731 XODD= 1.479462
N= 2.490 A= 1.577973 B= 1.267449 XEVEN= 0.730183 XODD= 1.460367
N= 2.500 A= 1.581139 B= 1.264911 XEVEN= 0.720759 XODD= 1.441518
N= 2.510 A= 1.584298 B= 1.262389 XEVEN= 0.711456 XODD= 1.422911
N= 2.520 A= 1.587451 B= 1.259882 XEVEN= 0.702270 XODD= 1.404540
N= 2.530 A= 1.590597 B= 1.257389 XEVEN= 0.693201 XODD= 1.386402
N= 2.540 A= 1.593738 B= 1.254912 XEVEN= 0.684245 XODD= 1.368491
N= 2.550 A= 1.596872 B= 1.252449 XEVEN= 0.675401 XODD= 1.350802
N= 2.560 A= 1.600000 B= 1.250000 XEVEN= 0.666667 XODD= 1.333333
N= 2.570 A= 1.603122 B= 1.247566 XEVEN= 0.658039 XODD= 1.316079
N= 2.580 A= 1.606238 B= 1.245146 XEVEN= 0.649518 XODD= 1.299035
N= 2.590 A= 1.609348 B= 1.242740 XEVEN= 0.641099 XODD= 1.282198
N= 2.600 A= 1.612452 B= 1.240347 XEVEN= 0.632782 XODD= 1.265564
N= 2.610 A= 1.615549 B= 1.237969 XEVEN= 0.624565 XODD= 1.249130
N= 2.620 A= 1.618641 B= 1.235604 XEVEN= 0.616445 XODD= 1.232891
N= 2.630 A= 1.621727 B= 1.233253 XEVEN= 0.608422 XODD= 1.216843
N= 2.640 A= 1.624808 B= 1.230915 XEVEN= 0.600492 XODD= 1.200985
N= 2.650 A= 1.627882 B= 1.228590 XEVEN= 0.592656 XODD= 1.185312
N= 2.660 A= 1.630951 B= 1.226279 XEVEN= 0.584910 XODD= 1.169820
N= 2.670 A= 1.634013 B= 1.223980 XEVEN= 0.577254 XODD= 1.154507
N= 2.680 A= 1.637071 B= 1.221694 XEVEN= 0.569685 XODD= 1.139370
N= 2.690 A= 1.640122 B= 1.219422 XEVEN= 0.562202 XODD= 1.124405
N= 2.700 A= 1.643168 B= 1.217161 XEVEN= 0.554805 XODD= 1.109609
N= 2.710 A= 1.646208 B= 1.214913 XEVEN= 0.547490 XODD= 1.094980
N= 2.720 A= 1.649242 B= 1.212678 XEVEN= 0.540257 XODD= 1.080514
N= 2.730 A= 1.652271 B= 1.210455 XEVEN= 0.533105 XODD= 1.066209
N= 2.740 A= 1.655295 B= 1.208244 XEVEN= 0.526031 XODD= 1.052063
N= 2.750 A= 1.658312 B= 1.206045 XEVEN= 0.519036 XODD= 1.038071
N= 2.760 A= 1.661325 B= 1.203859 XEVEN= 0.512116 XODD= 1.024233
N= 2.770 A= 1.664332 B= 1.201684 XEVEN= 0.505272 XODD= 1.010544
N= 2.780 A= 1.667333 B= 1.199520 XEVEN= 0.498502 XODD= 0.997004
N= 2.790 A= 1.670329 B= 1.197369 XEVEN= 0.491804 XODD= 0.983608
N= 2.800 A= 1.673320 B= 1.195229 XEVEN= 0.485178 XODD= 0.970356
N= 2.810 A= 1.676305 B= 1.193100 XEVEN= 0.478622 XODD= 0.957244
N= 2.820 A= 1.679286 B= 1.190983 XEVEN= 0.472135 XODD= 0.944270
N= 2.830 A= 1.682260 B= 1.188877 XEVEN= 0.465716 XODD= 0.931432
N= 2.840 A= 1.685230 B= 1.186782 XEVEN= 0.459364 XODD= 0.918728
N= 2.850 A= 1.688194 B= 1.184698 XEVEN= 0.453078 XODD= 0.906156
N= 2.860 A= 1.691153 B= 1.182625 XEVEN= 0.446857 XODD= 0.893713
N= 2.870 A= 1.694107 B= 1.180563 XEVEN= 0.440699 XODD= 0.881398
N= 2.880 A= 1.697056 B= 1.178511 XEVEN= 0.434604 XODD= 0.869209
N= 2.890 A= 1.700000 B= 1.176471 XEVEN= 0.428571 XODD= 0.857143
N= 2.900 A= 1.702939 B= 1.174440 XEVEN= 0.422599 XODD= 0.845199
N= 2.910 A= 1.705872 B= 1.172421 XEVEN= 0.416687 XODD= 0.833374
N= 2.920 A= 1.708801 B= 1.170411 XEVEN= 0.410834 XODD= 0.821667
N= 2.930 A= 1.711724 B= 1.168412 XEVEN= 0.405038 XODD= 0.810077
N= 2.940 A= 1.714643 B= 1.166424 XEVEN= 0.399300 XODD= 0.798601
N= 2.950 A= 1.717556 B= 1.164445 XEVEN= 0.393619 XODD= 0.787237
N= 2.960 A= 1.720465 B= 1.162476 XEVEN= 0.387992 XODD= 0.775985
N= 2.970 A= 1.723369 B= 1.160518 XEVEN= 0.382421 XODD= 0.764841
N= 2.980 A= 1.726268 B= 1.158569 XEVEN= 0.376903 XODD= 0.753806
N= 2.990 A= 1.729162 B= 1.156630 XEVEN= 0.371438 XODD= 0.742876
N= 3.000 A= 1.732051 B= 1.154701 XEVEN= 0.366025 XODD= 0.732051
N= 3.010 A= 1.734935 B= 1.152781 XEVEN= 0.360664 XODD= 0.721328
N= 3.020 A= 1.737815 B= 1.150871 XEVEN= 0.355354 XODD= 0.710708
N= 3.030 A= 1.740690 B= 1.148970 XEVEN= 0.350093 XODD= 0.700187
N= 3.040 A= 1.743560 B= 1.147079 XEVEN= 0.344882 XODD= 0.689764
N= 3.050 A= 1.746425 B= 1.145197 XEVEN= 0.339719 XODD= 0.679439
N= 3.060 A= 1.749286 B= 1.143324 XEVEN= 0.334605 XODD= 0.669209
N= 3.070 A= 1.752142 B= 1.141460 XEVEN= 0.329537 XODD= 0.659074
N= 3.080 A= 1.754993 B= 1.139606 XEVEN= 0.324516 XODD= 0.649032
N= 3.090 A= 1.757840 B= 1.137760 XEVEN= 0.319540 XODD= 0.639081
N= 3.100 A= 1.760682 B= 1.135924 XEVEN= 0.314610 XODD= 0.629221
N= 3.110 A= 1.763519 B= 1.134096 XEVEN= 0.309725 XODD= 0.619449
N= 3.120 A= 1.766352 B= 1.132277 XEVEN= 0.304883 XODD= 0.609766
N= 3.130 A= 1.769181 B= 1.130467 XEVEN= 0.300085 XODD= 0.600170
N= 3.140 A= 1.772005 B= 1.128665 XEVEN= 0.295329 XODD= 0.590658
N= 3.150 A= 1.774824 B= 1.126872 XEVEN= 0.290616 XODD= 0.581232
N= 3.160 A= 1.777639 B= 1.125088 XEVEN= 0.285944 XODD= 0.571888
N= 3.170 A= 1.780449 B= 1.123312 XEVEN= 0.281313 XODD= 0.562626
N= 3.180 A= 1.783255 B= 1.121544 XEVEN= 0.276723 XODD= 0.553445
N= 3.190 A= 1.786057 B= 1.119785 XEVEN= 0.272172 XODD= 0.544344
N= 3.200 A= 1.788854 B= 1.118034 XEVEN= 0.267661 XODD= 0.535322
N= 3.210 A= 1.791647 B= 1.116291 XEVEN= 0.263189 XODD= 0.526378
N= 3.220 A= 1.794436 B= 1.114556 XEVEN= 0.258755 XODD= 0.517510
N= 3.230 A= 1.797220 B= 1.112830 XEVEN= 0.254359 XODD= 0.508718
N= 3.240 A= 1.800000 B= 1.111111 XEVEN= 0.250000 XODD= 0.500000
N= 3.250 A= 1.802776 B= 1.109400 XEVEN= 0.245678 XODD= 0.491356
N= 3.260 A= 1.805547 B= 1.107698 XEVEN= 0.241392 XODD= 0.482785
N= 3.270 A= 1.808314 B= 1.106003 XEVEN= 0.237143 XODD= 0.474286
N= 3.280 A= 1.811077 B= 1.104315 XEVEN= 0.232929 XODD= 0.465857
N= 3.290 A= 1.813836 B= 1.102636 XEVEN= 0.228749 XODD= 0.457498
N= 3.300 A= 1.816590 B= 1.100964 XEVEN= 0.224604 XODD= 0.449209
N= 3.310 A= 1.819341 B= 1.099299 XEVEN= 0.220494 XODD= 0.440987
N= 3.320 A= 1.822087 B= 1.097643 XEVEN= 0.216417 XODD= 0.432833
N= 3.330 A= 1.824829 B= 1.095993 XEVEN= 0.212373 XODD= 0.424746
N= 3.340 A= 1.827567 B= 1.094351 XEVEN= 0.208362 XODD= 0.416724
N= 3.350 A= 1.830301 B= 1.092717 XEVEN= 0.204383 XODD= 0.408766
N= 3.360 A= 1.833030 B= 1.091089 XEVEN= 0.200437 XODD= 0.400873
N= 3.370 A= 1.835756 B= 1.089469 XEVEN= 0.196521 XODD= 0.393043
N= 3.380 A= 1.838478 B= 1.087857 XEVEN= 0.192638 XODD= 0.385275
N= 3.390 A= 1.841195 B= 1.086251 XEVEN= 0.188785 XODD= 0.377569
N= 3.400 A= 1.843909 B= 1.084652 XEVEN= 0.184962 XODD= 0.369924
N= 3.410 A= 1.846619 B= 1.083061 XEVEN= 0.181170 XODD= 0.362339
N= 3.420 A= 1.849324 B= 1.081476 XEVEN= 0.177407 XODD= 0.354813
N= 3.430 A= 1.852026 B= 1.079898 XEVEN= 0.173673 XODD= 0.347346
N= 3.440 A= 1.854724 B= 1.078328 XEVEN= 0.169969 XODD= 0.339937
N= 3.450 A= 1.857418 B= 1.076764 XEVEN= 0.166293 XODD= 0.332586
N= 3.460 A= 1.860108 B= 1.075207 XEVEN= 0.162645 XODD= 0.325291
N= 3.470 A= 1.862794 B= 1.073656 XEVEN= 0.159026 XODD= 0.318051
N= 3.480 A= 1.865476 B= 1.072113 XEVEN= 0.155434 XODD= 0.310868
N= 3.490 A= 1.868154 B= 1.070575 XEVEN= 0.151869 XODD= 0.303738
N= 3.500 A= 1.870829 B= 1.069045 XEVEN= 0.148331 XODD= 0.296663
N= 3.510 A= 1.873499 B= 1.067521 XEVEN= 0.144820 XODD= 0.289641
N= 3.520 A= 1.876166 B= 1.066004 XEVEN= 0.141336 XODD= 0.282672
N= 3.530 A= 1.878829 B= 1.064493 XEVEN= 0.137877 XODD= 0.275754
N= 3.540 A= 1.881489 B= 1.062988 XEVEN= 0.134444 XODD= 0.268889
N= 3.550 A= 1.884144 B= 1.061490 XEVEN= 0.131037 XODD= 0.262074
N= 3.560 A= 1.886796 B= 1.059998 XEVEN= 0.127655 XODD= 0.255310
N= 3.570 A= 1.889444 B= 1.058512 XEVEN= 0.124297 XODD= 0.248595
N= 3.580 A= 1.892089 B= 1.057033 XEVEN= 0.120965 XODD= 0.241929
N= 3.590 A= 1.894730 B= 1.055560 XEVEN= 0.117656 XODD= 0.235312
N= 3.600 A= 1.897367 B= 1.054093 XEVEN= 0.114372 XODD= 0.228743
N= 3.610 A= 1.900000 B= 1.052632 XEVEN= 0.111111 XODD= 0.222222
N= 3.620 A= 1.902630 B= 1.051177 XEVEN= 0.107874 XODD= 0.215748
N= 3.630 A= 1.905256 B= 1.049728 XEVEN= 0.104660 XODD= 0.209320
N= 3.640 A= 1.907878 B= 1.048285 XEVEN= 0.101469 XODD= 0.202938
N= 3.650 A= 1.910497 B= 1.046848 XEVEN= 0.098301 XODD= 0.196602
N= 3.660 A= 1.913113 B= 1.045417 XEVEN= 0.095155 XODD= 0.190310
N= 3.670 A= 1.915724 B= 1.043992 XEVEN= 0.092032 XODD= 0.184063
N= 3.680 A= 1.918333 B= 1.042572 XEVEN= 0.088930 XODD= 0.177860
N= 3.690 A= 1.920937 B= 1.041158 XEVEN= 0.085850 XODD= 0.171701
N= 3.700 A= 1.923538 B= 1.039750 XEVEN= 0.082792 XODD= 0.165584
N= 3.710 A= 1.926136 B= 1.038348 XEVEN= 0.079755 XODD= 0.159510
N= 3.720 A= 1.928730 B= 1.036952 XEVEN= 0.076739 XODD= 0.153478
N= 3.730 A= 1.931321 B= 1.035561 XEVEN= 0.073744 XODD= 0.147488
N= 3.740 A= 1.933908 B= 1.034175 XEVEN= 0.070769 XODD= 0.141539
N= 3.750 A= 1.936492 B= 1.032796 XEVEN= 0.067815 XODD= 0.135630
N= 3.760 A= 1.939072 B= 1.031421 XEVEN= 0.064881 XODD= 0.129762
N= 3.770 A= 1.941649 B= 1.030052 XEVEN= 0.061967 XODD= 0.123934
N= 3.780 A= 1.944222 B= 1.028689 XEVEN= 0.059073 XODD= 0.118145
N= 3.790 A= 1.946792 B= 1.027331 XEVEN= 0.056198 XODD= 0.112396
N= 3.800 A= 1.949359 B= 1.025978 XEVEN= 0.053342 XODD= 0.106685
N= 3.810 A= 1.951922 B= 1.024631 XEVEN= 0.050506 XODD= 0.101012
N= 3.820 A= 1.954482 B= 1.023289 XEVEN= 0.047689 XODD= 0.095377
N= 3.830 A= 1.957039 B= 1.021952 XEVEN= 0.044890 XODD= 0.089780
N= 3.840 A= 1.959592 B= 1.020621 XEVEN= 0.042110 XODD= 0.084220
N= 3.850 A= 1.962142 B= 1.019294 XEVEN= 0.039348 XODD= 0.078696
N= 3.860 A= 1.964688 B= 1.017973 XEVEN= 0.036604 XODD= 0.073209
N= 3.870 A= 1.967232 B= 1.016657 XEVEN= 0.033879 XODD= 0.067757
N= 3.880 A= 1.969772 B= 1.015346 XEVEN= 0.031171 XODD= 0.062341
N= 3.890 A= 1.972308 B= 1.014040 XEVEN= 0.028480 XODD= 0.056961
N= 3.900 A= 1.974842 B= 1.012739 XEVEN= 0.025807 XODD= 0.051615
N= 3.910 A= 1.977372 B= 1.011443 XEVEN= 0.023152 XODD= 0.046304
N= 3.920 A= 1.979899 B= 1.010153 XEVEN= 0.020513 XODD= 0.041027
N= 3.930 A= 1.982423 B= 1.008867 XEVEN= 0.017892 XODD= 0.035783
N= 3.940 A= 1.984943 B= 1.007585 XEVEN= 0.015287 XODD= 0.030574
N= 3.950 A= 1.987461 B= 1.006309 XEVEN= 0.012699 XODD= 0.025397
N= 3.960 A= 1.989975 B= 1.005038 XEVEN= 0.010127 XODD= 0.020253
N= 3.970 A= 1.992486 B= 1.003771 XEVEN= 0.007571 XODD= 0.015142
N= 3.980 A= 1.994994 B= 1.002509 XEVEN= 0.005031 XODD= 0.010063
N= 3.990 A= 1.997498 B= 1.001252 XEVEN= 0.002508 XODD= 0.005016
N= 4.000 A= 2.000000 B= 1.000000 XEVEN= 1.000000 XODD= 2.000000
APPENDIX II (program that produces data)
IMPLICIT INTEGER*4(A-Z)
PARAMETER XMAX=50
INTEGER X(0:XMAX)
DOUBLE PRECISION A,B,XODD,XEVEN,DSQRT
DO 10 N=100,400
A=DSQRT(DFLOAT(N)/100.0)
B=2.0D0/A
X(0)=1
XODD=0.0
XEVEN=0.0
DO 2 I=1,XMAX
IF (MOD(I,2).EQ.1) THEN
X(I)=A*X(I-1)+A/2
IF (I.NE.1) THEN
D=X(I)-2*X(I-2)
IF (D.NE.0 .AND. D.NE.1) TYPE 101, N/100.0,I,D,X(I)
EXP=-( (I-1)/2 )
XEVEN=XEVEN+D*2.0D0**EXP
END IF
ELSE
X(I)=B*X(I-1)+B/2
D=X(I)-2*X(I-2)
IF (D.NE.0 .AND. D.NE.1) TYPE 101, N/100.0,I,D,X(I)
EXP=-( (I-1)/2 )
XODD=XODD+D*2.0D0**EXP
END IF
101 FORMAT(' N=',F6.3,' I=',I3,' D=',I3,' X(I)=',I10)
2 CONTINUE
TYPE 100, N/100.0,A,B,XEVEN,XODD
100 FORMAT(' N=',F6.3,' A=',F12.6,' B=',F12.6,
X ' XEVEN=',F12.6,' XODD=',F12.6)
10 CONTINUE
END
|
| 67.10 | | TURTLE::GILBERT | | Wed Aug 01 1984 15:28 | 31 |
| Can someone apply Vosbury's technique to prove Stan's conjectures?
Actually, a stronger result could also allow for other than "a/2"
in the expression [ ax + a/2 ], and other than "2" in b = 2/a.
Instead of defining the sums over every other subscript, it could
consider sums over every third subscript (or 4th, ...).
n
Part of the proof requires showing that if x = [ B y ], then
n
[ ax + c ] = [ B z ], with a,B,z > 0. This is equivalent to
n n n n n n
B z - { B z } <= a ( B y - { B y } ) + c < B z - { B z } + 1
n
We require that z = ay (necessary for large B ). This gives:
n n
0 <= { B ay } - a { B y } + c < 1
n n
We choose { B ay } = { B y } = w, since otherwise the expression will vary too
much (although there may be very interesting cases for which this choice is not
necessary). This yields 0 <= w - aw + c < 1. If we assume that 0 < w < 1
are the best bounds on w (although any integer B and rational y give a counter-
example, as do many transcendental numbers), then this gives 0 < a - c < 1,
and 0 < c < 1.
Note that these inequalities, together with c = a/2, explain Stan's observation
about 1 < a < 2.
n n
The requirement that { B ay } = { B y } is perhaps the most interesting.
Note that B an integer, a = sqrt(k) and y = (1 + m sqrt(k)) are solutions.
- Gilbert
|
| 67.11 | | TOOLS::STAN | | Sun Dec 15 1985 22:58 | 3 |
| Gilbert and I have proven the conjecture in .9.
Contact either of us if you would like to see the proof.
|
| 67.12 | | TOOLS::STAN | | Sun Dec 15 1985 23:03 | 13 |
| In .9 I noticed that if d(n)=x(n)-2x(n-2), then d(n) was always either
0 or 1 if a was between 1 and 2.
I recently investigated other values of a.
If 2<a<4, then d apparently is always in the set {-1,0,1,2}.
If 4<a<6, then d apparently is always in the set {-2,-1,0,1,2,3}.
etc.
That is, if m=floor(a/2), and a>1, then d appears to always lie in the
interval [-m,m+1]. I have no proof of this yet.
|