Theorem
The Collatz conjecture states
that for any positive integer N, the following
transformational rules will if applied to N result in a set
of numbers whose trajectory ends with 1.
The simplest trajectory
demonstrating the rules is that of N = 1.
Rule 1. If the number is
odd times by 3 and add 1
Rule 2. If the number is even
divide by 2
Rule 3. Apply the rules again to
the product produced.
Thus for N = 1, the
trajectory is 1 -> 4 (1*3+1) -> 2 (4/2) -> 1
(2/2).
Every positive integer which has
been tested eventually reaches 1 under the Collatz map. (Equivalently, no
counterexample to the Collatz conjecture has been shown to exist.)
However there is no theoretical certainty that this is so.
The following argument by
contradiction, if strong, demonstrates the Collatz conjecture to be true, and
subject to peer review, suggests extensive boundary conditions on the existence
of non-Collatz numbers in general and the ‘lowest’ non-Collatz number specifically
which could be the basis for a stronger formulation if needed.
Definitions and Setup
Let a c‑number be
a positive integer whose Collatz trajectory reaches 1. Let a z‑number be
a positive integer whose Collatz trajectory does not reach 1.
Such a number if it existed might have a looping trajectory composed entirely
of z-numbers, or increase to infinity, but the curve on which it would be the
lowest point would exist entirely of z- numbers. By definition a c-number
cannot give rise to a z-number, as by doing so its trajectory would cease to
reach 1. The converse is also a z-number cannot give rise to a c-number,
as by doing so its trajectory would reach 1.
Assume for contradiction that z‑numbers
exist.
A lowest or least z-number must
therefore exist.
Let Z(0) denote the least z‑number.
Lemma 1 — Z(0) must be odd
If Z(0) were even, then its Collatz
product would be Z(0)/2, which is smaller than Z(0). Since Z(0) is assumed to
be the least z‑number,
this would be a contradiction. Thus
Z(0) is odd.
Lemma 2 — 2Z(0) is the
least even z‑number
Observe that 2Z(0) is even and its Collatz product
would be is Z(0). Z(0) is a z-number, therefore 2Z(0) is a
z-number.
Suppose there were an even z‑number E
smaller than 2Z(0). Then E's Collatz trajectory
would give a z- number smaller than
Z(0) contradicting its the minimality
Lemma 3 — nZ(0) where n is an even multiple >2 is
the least even z‑number having the property of producing an odd number in its
Collatz trajectory after a specific number of steps s greater
than 0.
This follows from Lemma
2. 2Z(0) is the smallest even z‑number. Observe that it
also has the definable property of being the smallest even z-number to
immediately give rise to an odd z-number. If a smaller even z-number did
this its trajectory would give rise to a smaller z-number than 2Z(0) which
would be a contradiction.
This chain can be projected back
thus:
Z(0) is the smallest odd z‑number.
2Z(0) is the smallest even z‑number, which produces an
odd z-number by 1 direct transform.
I will define this as an even
z-number of type 0 as the steps S between 2. Types
1,2,3….would then be definable as below:
Type 1
4Z(0) is the smallest even z‑number, which produces 1
even z-number, then an odd z-number.
Type 2
8Z(0) is the smallest even z‑number, which produces 2
even z-numbers, then an odd z-number.
Type 3
16Z(0) is the smallest even z‑number, which produces 3
even z-numbers, then an odd z-number.
…..etc
This precusor chain can, without
inherent contradiction be extended to infinity. Indeed
it is arguable that the
existence of a lowest odd z-number, requires such a chain of even
z-numbers to which we can assign
negative suffixes as the approach the lowest odd z-number,
This produces the necessary
chain -
…..Z(-4) Z(-3) Z(-2)
Z(-1) Z(0) 3Z(0)+1………
…...16Z(0) 8Z(0) 4Z(0)
2Z(0) Z(0) Z(1)………….
Lemma 4 — the third transform of Z0 must
be odd.
As an odd number Z0 gives the even number 3*Z0+1 as its first transform
Z(1). It’s second transform Z(2) is
(3*Z(0)+1)/2 or 1.5Z(0)+0.5. This number Z(2) must be odd or
it would produce (for any starting
Z(0) > 1, an even number <
2Z(0) in contradiction of Lemma 2.
Lemma
5 — The above requirements impose the following consequent restraints on the z-number
placement around Z(0) and hence its possible Collatz trajectory.
5a No even number in the
interval (Z(0)+1, 2Z(0)-2) is a z‑number.
If an even z-number in the range
above exists its Collatz trajectory includes at least one z-number < Z(0) This
would be contradictory.
Furthermore for each type of
even z-number the range above necessarily increases, so no even z-number of
type 1 can exist in the range (Z(0)+1,
4Z(0)-2)
No even z-numbers of type 2 can
exist in the range (Z(0)+1,
8Z(0)-2).
Because computer checking has
extended the known set of Collatz compliant numbers to on the order of 2.36×1021
it can be seen that the likely
number of even integers in the above interval must be at the lowest on the
order of 2.36x1021 / 4.
This is important because in
order as we will see to be valid Z0 ‘s trajectory will need to avoid intersection
not only with that set of even integers but with every value in their Collatz
trajectories, as Lemma 6 will set out.
5b No even number in the
precursor chain of an odd c-number is a z‑number.
Every c-number lower than Z(0)
can be thought of as having a precursor chain just as Z(0) has. For example consider the odd
Numbers immediately below Z(0)
and their precusors
…..16(Z(0)-2)…..8(Z(0)-2)…..4(Z0-2)….2(Z0-2)…,Z(0)-2
…..16(Z(0)-4)…..8(Z(0)-4)…..4(Z0-4)….2(Z0-4)…,Z(0)-4
…..16(Z(0)-8)…..8(Z(0)-8)…..4(Z0-8)….2(Z0-8)…,Z(4)-8
…..16(Z(0)-10)…..8(Z(0)-10)…..4(Z0-10)….2(Z0-10)…,Z(0)-10….
5c Every other odd number higher
than Z(0) [at least] is a c-number.
By Lemma 4 Z(0)’s second transform is odd. Consider the transform paths of Z(0)+2, and
Z(0)+4 in parallel to Z(0).
Z(0) 3Z(0)+1 1.5Z(0)+0.5 pattern odd even
(by Lemma 4 ) odd
Z(0)+2 3(Z0+2)+1 1.5(Z(0)+2)+0.5 odd even even
Z(0)+4 3(Z0+4)+1 1.5(Z(0)+4)+0.5 odd even odd
Z(0)+6 3(Z0+4)+1 1.5(Z(0)+6)+0.5) odd even even
The bold, alternating odd
numbers cannot be z-numbers because their second
transform will be even, giving a value
<2Z0 in contradiction of
Lemma 2. Because of the large magnitude
of any possible Z(0), the +values make no impact on the
effect of this which can be
considered to continue to Z(0)+n. This imposes no requirement on non-bold
numbers as to whether
or not they are c- or z- type
numbers.
Lemma 6 — Any Collatz trajectory
starting Z(0) must to be possible, never intersect with any of the c-numbers
required by Lemma 5, nor any number produced by any of their trajectories, or
their precursor chains.
Effect of Lemma 5a constraint.
Trajectories can be uniquely
identified by their sequence of odd-even-numbers. I will 'name'
trajectories by assigning 1 to odd and 0 to even, giving a value.
For the first 14 steps (including the starting number) this gives 582 possible
trajectories from an odd start. This gives paths from:
100000000000000 to
101010101010101
It can be shown exhaustively by
analysis that for the first 14 steps from any Z0 the set of possible
odd-even-etc trajectories results for all produce:
1) in 552 of 582 cases, a simple failure due to the necessary
creation of an even value less than 2Z(0).
2) in 30 of the 582, no failure due to
Lemma 5a constraints within 15 steps, however methodologies for
testing constraints from Lemma 5b and 5c
are under construction, and will be added to this paper.
Example of Lemma 5a constraint:
For example the trajectory 101010101000001 where 1 = odd, and 0 = even results
in a number of the form 0.474609* S+0.412109 at step 15.
If S = Z0, the trajectory at step 15 is <2Z0,
thus S for that trajectory cannot consistantly = Z0
Full listing of 15 step analysis
for possible Z(0) path constraints from Lemma 5a:
Trajectories which if followed
by illustrative c-numbers lead to coalescence with trajectories in the 2x S
range after 14 steps.
|
Ref |
Steps 1= odd, 0 =even |
resultant value at step14 from SEED S |
Notes |
|
|
1 |
100000000000000 |
0.0003662109375*S+0.0001220703125 |
Too low at step2 for Z0 validity |
|
|
2 |
100000000000001 |
0.0003662109375*S+0.0001220703125 |
Too low at step2 for Z0 validity |
|
|
3 |
100000000000010 |
0.002197265625*S+1.000732421875 |
Too low at step2 for Z0 validity |
|
|
4 |
100000000000100 |
0.002197265625*S+0.500732421875 |
Too low at step2 for Z0 validity |
|
|
5 |
100000000000101 |
0.002197265625*S+0.500732421875 |
Too low at step2 for Z0 validity |
|
|
6 |
100000000001000 |
0.002197265625*S+0.250732421875 |
Too low at step2 for Z0 validity |
|
|
7 |
100000000001001 |
0.002197265625*S+0.250732421875 |
Too low at step2 for Z0 validity |
|
|
8 |
100000000001010 |
0.01318359375*S+2.50439453125 |
Too low at step2 for Z0 validity |
|
|
9 |
100000000010000 |
0.002197265625*S+0.125732421875 |
Too low at step2 for Z0 validity |
|
|
10 |
100000000010001 |
0.002197265625*S+0.125732421875 |
Too low at step2 for Z0 validity |
|
|
11 |
100000000010010 |
0.01318359375*S+1.75439453125 |
Too low at step2 for Z0 validity |
|
|
12 |
100000000010100 |
0.01318359375*S+1.25439453125 |
Too low at step2 for Z0 validity |
|
|
13 |
100000000010101 |
0.01318359375*S+1.25439453125 |
Too low at step2 for Z0 validity |
|
|
14 |
100000000100000 |
0.002197265625*S+0.063232421875 |
Too low at step2 for Z0 validity |
|
|
15 |
100000000100001 |
0.002197265625*S+0.063232421875 |
Too low at step2 for Z0 validity |
|
|
16 |
100000000100010 |
0.01318359375*S+1.37939453125 |
Too low at step2 for Z0 validity |
|
|
17 |
100000000100100 |
0.01318359375*S+0.87939453125 |
Too low at step2 for Z0 validity |
|
|
18 |
100000000100101 |
0.01318359375*S+0.87939453125 |
Too low at step2 for Z0 validity |
|
|
19 |
100000000101000 |
0.01318359375*S+0.62939453125 |
Too low at step2 for Z0 validity |
|
|
20 |
100000000101001 |
0.01318359375*S+0.62939453125 |
Too low at step2 for Z0 validity |
|
|
21 |
100000000101010 |
0.0791015625*S+4.7763671875 |
Too low at step2 for Z0 validity |
|
|
22 |
100000001000000 |
0.002197265625*S+0.031982421875 |
Too low at step2 for Z0 validity |
|
|
23 |
100000001000001 |
0.002197265625*S+0.031982421875 |
Too low at step2 for Z0 validity |
|
|
24 |
100000001000010 |
0.01318359375*S+1.19189453125 |
Too low at step2 for Z0 validity |
|
|
25 |
100000001000100 |
0.01318359375*S+0.69189453125 |
Too low at step2 for Z0 validity |
|
|
26 |
100000001000101 |
0.01318359375*S+0.69189453125 |
Too low at step2 for Z0 validity |
|
|
27 |
100000001001000 |
0.01318359375*S+0.44189453125 |
Too low at step2 for Z0 validity |
|
|
28 |
100000001001001 |
0.01318359375*S+0.44189453125 |
Too low at step2 for Z0 validity |
|
|
29 |
100000001001010 |
0.0791015625*S+3.6513671875 |
Too low at step2 for Z0 validity |
|
|
30 |
100000001010000 |
0.01318359375*S+0.31689453125 |
Too low at step2 for Z0 validity |
|
|
31 |
100000001010001 |
0.01318359375*S+0.31689453125 |
Too low at step2 for Z0 validity |
|
|
32 |
100000001010010 |
0.0791015625*S+2.9013671875 |
Too low at step2 for Z0 validity |
|
|
33 |
100000001010100 |
0.0791015625*S+2.4013671875 |
Too low at step2 for Z0 validity |
|
|
34 |
100000001010101 |
0.0791015625*S+2.4013671875 |
Too low at step2 for Z0 validity |
|
|
35 |
100000010000000 |
0.002197265625*S+0.016357421875 |
Too low at step2 for Z0 validity |
|
|
36 |
100000010000001 |
0.002197265625*S+0.016357421875 |
Too low at step2 for Z0 validity |
|
|
37 |
100000010000010 |
0.01318359375*S+1.09814453125 |
Too low at step2 for Z0 validity |
|
|
38 |
100000010000100 |
0.01318359375*S+0.59814453125 |
Too low at step2 for Z0 validity |
|
|
39 |
100000010000101 |
0.01318359375*S+0.59814453125 |
Too low at step2 for Z0 validity |
|
|
40 |
100000010001000 |
0.01318359375*S+0.34814453125 |
Too low at step2 for Z0 validity |
|
|
41 |
100000010001010 |
0.0791015625*S+3.0888671875 |
Too low at step2 for Z0 validity |
|
|
42 |
100000010010000 |
0.01318359375*S+0.22314453125 |
Too low at step2 for Z0 validity |
|
|
43 |
100000010010001 |
0.01318359375*S+0.22314453125 |
Too low at step2 for Z0 validity |
|
|
44 |
100000010010010 |
0.0791015625*S+2.3388671875 |
Too low at step2 for Z0 validity |
|
|
45 |
100000010010100 |
0.0791015625*S+1.8388671875 |
Too low at step2 for Z0 validity |
|
|
46 |
100000010010101 |
0.0791015625*S+1.8388671875 |
Too low at step2 for Z0 validity |
|
|
47 |
100000010100000 |
0.01318359375*S+0.16064453125 |
Too low at step2 for Z0 validity |
|
|
48 |
100000010100001 |
0.01318359375*S+0.16064453125 |
Too low at step2 for Z0 validity |
|
|
49 |
100000010100010 |
0.0791015625*S+1.9638671875 |
Too low at step2 for Z0 validity |
|
|
50 |
100000010100101 |
0.0791015625*S+1.4638671875 |
Too low at step2 for Z0 validity |
|
|
51 |
100000010101000 |
0.0791015625*S+1.2138671875 |
Too low at step2 for Z0 validity |
|
|
52 |
100000010101001 |
0.0791015625*S+1.2138671875 |
Too low at step2 for Z0 validity |
|
|
53 |
100000010101010 |
0.474609375*S+8.283203125 |
Too low at step2 for Z0 validity |
|
|
54 |
100000100000000 |
0.002197265625*S+0.008544921875 |
Too low at step2 for Z0 validity |
|
|
55 |
100000100000001 |
0.002197265625*S+0.008544921875 |
Too low at step2 for Z0 validity |
|
|
56 |
100000100000010 |
0.01318359375*S+1.05126953125 |
Too low at step2 for Z0 validity |
|
|
57 |
100000100000100 |
0.01318359375*S+0.55126953125 |
Too low at step2 for Z0 validity |
|
|
58 |
100000100000101 |
0.01318359375*S+0.55126953125 |
Too low at step2 for Z0 validity |
|
|
59 |
100000100001000 |
0.01318359375*S+0.30126953125 |
Too low at step2 for Z0 validity |
|
|
60 |
100000100001001 |
0.01318359375*S+0.30126953125 |
Too low at step2 for Z0 validity |
|
|
61 |
100000100001010 |
0.0791015625*S+2.8076171875 |
Too low at step2 for Z0 validity |
|
|
62 |
100000100010000 |
0.01318359375*S+0.17626953125 |
Too low at step2 for Z0 validity |
|
|
63 |
100000100010001 |
0.01318359375*S+0.17626953125 |
Too low at step2 for Z0 validity |
|
|
64 |
100000100010010 |
0.0791015625*S+2.0576171875 |
Too low at step2 for Z0 validity |
|
|
65 |
100000100010100 |
0.0791015625*S+1.5576171875 |
Too low at step2 for Z0 validity |
|
|
66 |
100000100010101 |
0.0791015625*S+1.5576171875 |
Too low at step2 for Z0 validity |
|
|
67 |
100000100100000 |
0.01318359375*S+0.11376953125 |
Too low at step2 for Z0 validity |
|
|
68 |
100000100100001 |
0.01318359375*S+0.11376953125 |
Too low at step2 for Z0 validity |
|
|
69 |
100000100100010 |
0.0791015625*S+1.6826171875 |
Too low at step2 for Z0 validity |
|
|
70 |
100000100100100 |
0.0791015625*S+1.1826171875 |
Too low at step2 for Z0 validity |
|
|
71 |
100000100100101 |
0.0791015625*S+1.1826171875 |
Too low at step2 for Z0 validity |
|
|
72 |
100000100101000 |
0.0791015625*S+0.9326171875 |
Too low at step2 for Z0 validity |
|
|
73 |
100000100101001 |
0.0791015625*S+0.9326171875 |
Too low at step2 for Z0 validity |
|
|
74 |
100000100101010 |
0.474609375*S+6.595703125 |
Too low at step2 for Z0 validity |
|
|
75 |
100000101000000 |
0.01318359375*S+0.08251953125 |
Too low at step2 for Z0 validity |
|
|
76 |
100000101000001 |
0.01318359375*S+0.08251953125 |
Too low at step2 for Z0 validity |
|
|
77 |
100000101000010 |
0.0791015625*S+1.4951171875 |
Too low at step2 for Z0 validity |
|
|
78 |
100000101000100 |
0.0791015625*S+0.9951171875 |
Too low at step2 for Z0 validity |
|
|
79 |
100000101000101 |
0.0791015625*S+0.9951171875 |
Too low at step2 for Z0 validity |
|
|
80 |
100000101001000 |
0.0791015625*S+0.7451171875 |
Too low at step2 for Z0 validity |
|
|
81 |
100000101001001 |
0.0791015625*S+0.7451171875 |
Too low at step2 for Z0 validity |
|
|
82 |
100000101010000 |
0.0791015625*S+0.6201171875 |
Too low at step2 for Z0 validity |
|
|
83 |
100000101010001 |
0.0791015625*S+0.6201171875 |
Too low at step2 for Z0 validity |
|
|
84 |
100000101010010 |
0.474609375*S+4.720703125 |
Too low at step2 for Z0 validity |
|
|
85 |
100000101010100 |
0.474609375*S+4.220703125 |
Too low at step2 for Z0 validity |
|
|
86 |
100000101010101 |
0.474609375*S+4.220703125 |
Too low at step2 for Z0 validity |
|
|
87 |
100001000000001 |
0.002197265625*S+0.004638671875 |
Too low at step2 for Z0 validity |
|
|
88 |
100001000000010 |
0.01318359375*S+1.02783203125 |
Too low at step2 for Z0 validity |
|
|
89 |
100001000000100 |
0.01318359375*S+0.52783203125 |
Too low at step2 for Z0 validity |
|
|
90 |
100001000000101 |
0.01318359375*S+0.52783203125 |
Too low at step2 for Z0 validity |
|
|
91 |
100001000001000 |
0.01318359375*S+0.27783203125 |
Too low at step2 for Z0 validity |
|
|
92 |
100001000001001 |
0.01318359375*S+0.27783203125 |
Too low at step2 for Z0 validity |
|
|
93 |
100001000001010 |
0.0791015625*S+2.6669921875 |
Too low at step2 for Z0 validity |
|
|
94 |
100001000010000 |
0.01318359375*S+0.15283203125 |
Too low at step2 for Z0 validity |
|
|
95 |
100001000010001 |
0.01318359375*S+0.15283203125 |
Too low at step2 for Z0 validity |
|
|
96 |
100001000010010 |
0.0791015625*S+1.9169921875 |
Too low at step2 for Z0 validity |
|
|
97 |
100001000010100 |
0.0791015625*S+1.4169921875 |
Too low at step2 for Z0 validity |
|
|
98 |
100001000010101 |
0.0791015625*S+1.4169921875 |
Too low at step2 for Z0 validity |
|
|
99 |
100001000100000 |
0.01318359375*S+0.09033203125 |
Too low at step2 for Z0 validity |
|
|
100 |
100001000100001 |
0.01318359375*S+0.09033203125 |
Too low at step2 for Z0 validity |
|
|
101 |
100001000100010 |
0.0791015625*S+1.5419921875 |
Too low at step2 for Z0 validity |
|
|
102 |
100001000100100 |
0.0791015625*S+1.0419921875 |
Too low at step2 for Z0 validity |
|
|
103 |
100001000100101 |
0.0791015625*S+1.0419921875 |
Too low at step2 for Z0 validity |
|
|
104 |
100001000101000 |
0.0791015625*S+0.7919921875 |
Too low at step2 for Z0 validity |
|
|
105 |
100001000101001 |
0.0791015625*S+0.7919921875 |
Too low at step2 for Z0 validity |
|
|
106 |
100001000101010 |
0.474609375*S+5.751953125 |
Too low at step2 for Z0 validity |
|
|
107 |
100001001000000 |
0.01318359375*S+0.05908203125 |
Too low at step2 for Z0 validity |
|
|
108 |
100001001000001 |
0.01318359375*S+0.05908203125 |
Too low at step2 for Z0 validity |
|
|
109 |
100001001000010 |
0.0791015625*S+1.3544921875 |
Too low at step2 for Z0 validity |
|
|
110 |
100001001000100 |
0.0791015625*S+0.8544921875 |
Too low at step2 for Z0 validity |
|
|
111 |
100001001000101 |
0.0791015625*S+0.8544921875 |
Too low at step2 for Z0 validity |
|
|
112 |
100001001001000 |
0.0791015625*S+0.6044921875 |
Too low at step2 for Z0 validity |
|
|
113 |
100001001001001 |
0.0791015625*S+0.6044921875 |
Too low at step2 for Z0 validity |
|
|
114 |
100001001001010 |
0.474609375*S+4.626953125 |
Too low at step2 for Z0 validity |
|
|
115 |
100001001010000 |
0.0791015625*S+0.4794921875 |
Too low at step2 for Z0 validity |
|
|
116 |
100001001010001 |
0.0791015625*S+0.4794921875 |
Too low at step2 for Z0 validity |
|
|
117 |
100001001010010 |
0.474609375*S+3.876953125 |
Too low at step2 for Z0 validity |
|
|
118 |
100001001010100 |
0.474609375*S+3.376953125 |
Too low at step2 for Z0 validity |
|
|
119 |
100001001010101 |
0.474609375*S+3.376953125 |
Too low at step2 for Z0 validity |
|
|
120 |
100001010000000 |
0.01318359375*S+0.04345703125 |
Too low at step2 for Z0 validity |
|
|
121 |
100001010000001 |
0.01318359375*S+0.04345703125 |
Too low at step2 for Z0 validity |
|
|
122 |
100001010000010 |
0.0791015625*S+1.2607421875 |
Too low at step2 for Z0 validity |
|
|
123 |
100001010000100 |
0.0791015625*S+0.7607421875 |
Too low at step2 for Z0 validity |
|
|
124 |
100001010000101 |
0.0791015625*S+0.7607421875 |
Too low at step2 for Z0 validity |
|
|
125 |
100001010001000 |
0.0791015625*S+0.5107421875 |
Too low at step2 for Z0 validity |
|
|
126 |
100001010001001 |
0.0791015625*S+0.5107421875 |
Too low at step2 for Z0 validity |
|
|
127 |
100001010001010 |
0.474609375*S+4.064453125 |
Too low at step2 for Z0 validity |
|
|
128 |
100001010010000 |
0.0791015625*S+0.3857421875 |
Too low at step2 for Z0 validity |
|
|
129 |
100001010010001 |
0.0791015625*S+0.3857421875 |
Too low at step2 for Z0 validity |
|
|
130 |
100001010010010 |
0.474609375*S+3.314453125 |
Too low at step2 for Z0 validity |
|
|
131 |
100001010010100 |
0.474609375*S+2.814453125 |
Too low at step2 for Z0 validity |
|
|
132 |
100001010010101 |
0.474609375*S+2.814453125 |
Too low at step2 for Z0 validity |
|
|
133 |
100001010100000 |
0.0791015625*S+0.3232421875 |
Too low at step2 for Z0 validity |
|
|
134 |
100001010100001 |
0.0791015625*S+0.3232421875 |
Too low at step2 for Z0 validity |
|
|
135 |
100001010100010 |
0.474609375*S+2.939453125 |
Too low at step2 for Z0 validity |
|
|
136 |
100001010100100 |
0.474609375*S+2.439453125 |
Too low at step2 for Z0 validity |
|
|
137 |
100001010100101 |
0.474609375*S+2.439453125 |
Too low at step2 for Z0 validity |
|
|
138 |
100001010101000 |
0.474609375*S+2.189453125 |
Too low at step2 for Z0 validity |
|
|
139 |
100001010101001 |
0.474609375*S+2.189453125 |
Too low at step2 for Z0 validity |
|
|
140 |
100001010101010 |
2.84765625*S+14.13671875 |
Too low at step2 for Z0 validity |
|
|
141 |
100010000000000 |
0.002197265625*S+0.002685546875 |
Too low at step2 for Z0 validity |
|
|
142 |
100010000000001 |
0.002197265625*S+0.002685546875 |
Too low at step2 for Z0 validity |
|
|
143 |
100010000000010 |
0.01318359375*S+1.01611328125 |
Too low at step2 for Z0 validity |
|
|
144 |
100010000000100 |
0.01318359375*S+0.51611328125 |
Too low at step2 for Z0 validity |
|
|
145 |
100010000000101 |
0.01318359375*S+0.51611328125 |
Too low at step2 for Z0 validity |
|
|
146 |
100010000001000 |
0.01318359375*S+0.26611328125 |
Too low at step2 for Z0 validity |
|
|
147 |
100010000001001 |
0.01318359375*S+0.26611328125 |
Too low at step2 for Z0 validity |
|
|
148 |
100010000001010 |
0.0791015625*S+2.5966796875 |
Too low at step2 for Z0 validity |
|
|
149 |
100010000010000 |
0.01318359375*S+0.14111328125 |
Too low at step2 for Z0 validity |
|
|
150 |
100010000010001 |
0.01318359375*S+0.14111328125 |
Too low at step2 for Z0 validity |
|
|
151 |
100010000010010 |
0.0791015625*S+1.8466796875 |
Too low at step2 for Z0 validity |
|
|
152 |
100010000010100 |
0.0791015625*S+1.3466796875 |
Too low at step2 for Z0 validity |
|
|
153 |
100010000010101 |
0.0791015625*S+1.3466796875 |
Too low at step2 for Z0 validity |
|
|
154 |
100010000100000 |
0.01318359375*S+0.07861328125 |
Too low at step2 for Z0 validity |
|
|
155 |
100010000100001 |
0.01318359375*S+0.07861328125 |
Too low at step2 for Z0 validity |
|
|
156 |
100010000100010 |
0.0791015625*S+1.4716796875 |
Too low at step2 for Z0 validity |
|
|
157 |
100010000100100 |
0.0791015625*S+0.9716796875 |
Too low at step2 for Z0 validity |
|
|
158 |
100010000100101 |
0.0791015625*S+0.9716796875 |
Too low at step2 for Z0 validity |
|
|
159 |
100010000101001 |
0.0791015625*S+0.7216796875 |
Too low at step2 for Z0 validity |
|
|
160 |
100010000101010 |
0.474609375*S+5.330078125 |
Too low at step2 for Z0 validity |
|
|
161 |
100010001000000 |
0.01318359375*S+0.04736328125 |
Too low at step2 for Z0 validity |
|
|
162 |
100010001000001 |
0.01318359375*S+0.04736328125 |
Too low at step2 for Z0 validity |
|
|
163 |
100010001000010 |
0.0791015625*S+1.2841796875 |
Too low at step2 for Z0 validity |
|
|
164 |
100010001000100 |
0.0791015625*S+0.7841796875 |
Too low at step2 for Z0 validity |
|
|
165 |
100010001000101 |
0.0791015625*S+0.7841796875 |
Too low at step2 for Z0 validity |
|
|
166 |
100010001001000 |
0.0791015625*S+0.5341796875 |
Too low at step2 for Z0 validity |
|
|
167 |
100010001001001 |
0.0791015625*S+0.5341796875 |
Too low at step2 for Z0 validity |
|
|
168 |
100010001001010 |
0.474609375*S+4.205078125 |
Too low at step2 for Z0 validity |
|
|
169 |
100010001010000 |
0.0791015625*S+0.4091796875 |
Too low at step2 for Z0 validity |
|
|
170 |
100010001010001 |
0.0791015625*S+0.4091796875 |
Too low at step2 for Z0 validity |
|
|
171 |
100010001010010 |
0.474609375*S+3.455078125 |
Too low at step2 for Z0 validity |
|
|
172 |
100010001010100 |
0.474609375*S+2.955078125 |
Too low at step2 for Z0 validity |
|
|
173 |
100010001010101 |
0.474609375*S+2.955078125 |
Too low at step2 for Z0 validity |
|
|
174 |
100010010000000 |
0.01318359375*S+0.03173828125 |
Too low at step2 for Z0 validity |
|
|
175 |
100010010000001 |
0.01318359375*S+0.03173828125 |
Too low at step2 for Z0 validity |
|
|
176 |
100010010000010 |
0.0791015625*S+1.1904296875 |
Too low at step2 for Z0 validity |
|
|
177 |
100010010000100 |
0.0791015625*S+0.6904296875 |
Too low at step2 for Z0 validity |
|
|
178 |
100010010000101 |
0.0791015625*S+0.6904296875 |
Too low at step2 for Z0 validity |
|
|
179 |
100010010001000 |
0.0791015625*S+0.4404296875 |
Too low at step2 for Z0 validity |
|
|
180 |
100010010001001 |
0.0791015625*S+0.4404296875 |
Too low at step2 for Z0 validity |
|
|
181 |
100010010001010 |
0.474609375*S+3.642578125 |
Too low at step2 for Z0 validity |
|
|
182 |
100010010010000 |
0.0791015625*S+0.3154296875 |
Too low at step2 for Z0 validity |
|
|
183 |
100010010010001 |
0.0791015625*S+0.3154296875 |
Too low at step2 for Z0 validity |
|
|
184 |
100010010010010 |
0.474609375*S+2.892578125 |
Too low at step2 for Z0 validity |
|
|
185 |
100010010010100 |
0.474609375*S+2.392578125 |
Too low at step2 for Z0 validity |
|
|
186 |
100010010010101 |
0.474609375*S+2.392578125 |
Too low at step2 for Z0 validity |
|
|
187 |
100010010100000 |
0.0791015625*S+0.2529296875 |
Too low at step2 for Z0 validity |
|
|
188 |
100010010100001 |
0.0791015625*S+0.2529296875 |
Too low at step2 for Z0 validity |
|
|
189 |
100010010100010 |
0.474609375*S+2.517578125 |
Too low at step2 for Z0 validity |
|
|
190 |
100010010100100 |
0.474609375*S+2.017578125 |
Too low at step2 for Z0 validity |
|
|
191 |
100010010100101 |
0.474609375*S+2.017578125 |
Too low at step2 for Z0 validity |
|
|
192 |
100010010101000 |
0.474609375*S+1.767578125 |
Too low at step2 for Z0 validity |
|
|
193 |
100010010101010 |
2.84765625*S+11.60546875 |
Too low at step2 for Z0 validity |
|
|
194 |
100010100000000 |
0.01318359375*S+0.02392578125 |
Too low at step2 for Z0 validity |
|
|
195 |
100010100000001 |
0.01318359375*S+0.02392578125 |
Too low at step2 for Z0 validity |
|
|
196 |
100010100000010 |
0.0791015625*S+1.1435546875 |
Too low at step2 for Z0 validity |
|
|
197 |
100010100000100 |
0.0791015625*S+0.6435546875 |
Too low at step2 for Z0 validity |
|
|
198 |
100010100000101 |
0.0791015625*S+0.6435546875 |
Too low at step2 for Z0 validity |
|
|
199 |
100010100001000 |
0.0791015625*S+0.3935546875 |
Too low at step2 for Z0 validity |
|
|
200 |
100010100001001 |
0.0791015625*S+0.3935546875 |
Too low at step2 for Z0 validity |
|
|
201 |
100010100001010 |
0.474609375*S+3.361328125 |
Too low at step2 for Z0 validity |
|
|
202 |
100010100010000 |
0.0791015625*S+0.2685546875 |
Too low at step2 for Z0 validity |
|
|
203 |
100010100010001 |
0.0791015625*S+0.2685546875 |
Too low at step2 for Z0 validity |
|
|
204 |
100010100010010 |
0.474609375*S+2.611328125 |
Too low at step2 for Z0 validity |
|
|
205 |
100010100010100 |
0.474609375*S+2.111328125 |
Too low at step2 for Z0 validity |
|
|
206 |
100010100010101 |
0.474609375*S+2.111328125 |
Too low at step2 for Z0 validity |
|
|
207 |
100010100100000 |
0.0791015625*S+0.2060546875 |
Too low at step2 for Z0 validity |
|
|
208 |
100010100100001 |
0.0791015625*S+0.2060546875 |
Too low at step2 for Z0 validity |
|
|
209 |
100010100100010 |
0.474609375*S+2.236328125 |
Too low at step2 for Z0 validity |
|
|
210 |
100010100100100 |
0.474609375*S+1.736328125 |
Too low at step2 for Z0 validity |
|
|
211 |
100010100100101 |
0.474609375*S+1.736328125 |
Too low at step2 for Z0 validity |
|
|
212 |
100010100101000 |
0.474609375*S+1.486328125 |
Too low at step2 for Z0 validity |
|
|
213 |
100010100101001 |
0.474609375*S+1.486328125 |
Too low at step2 for Z0 validity |
|
|
214 |
100010100101010 |
2.84765625*S+9.91796875 |
Too low at step2 for Z0 validity |
|
|
215 |
100010101000000 |
0.0791015625*S+0.1748046875 |
Too low at step2 for Z0 validity |
|
|
216 |
100010101000001 |
0.0791015625*S+0.1748046875 |
Too low at step2 for Z0 validity |
|
|
217 |
100010101000010 |
0.474609375*S+2.048828125 |
Too low at step2 for Z0 validity |
|
|
218 |
100010101000100 |
0.474609375*S+1.548828125 |
Too low at step2 for Z0 validity |
|
|
219 |
100010101000101 |
0.474609375*S+1.548828125 |
Too low at step2 for Z0 validity |
|
|
220 |
100010101001000 |
0.474609375*S+1.298828125 |
Too low at step2 for Z0 validity |
|
|
221 |
100010101001001 |
0.474609375*S+1.298828125 |
Too low at step2 for Z0 validity |
|
|
222 |
100010101001010 |
2.84765625*S+8.79296875 |
Too low at step2 for Z0 validity |
|
|
223 |
100010101010000 |
0.474609375*S+1.173828125 |
Too low at step2 for Z0 validity |
|
|
224 |
100010101010001 |
0.474609375*S+1.173828125 |
Too low at step2 for Z0 validity |
|
|
225 |
100010101010010 |
2.84765625*S+8.04296875 |
Too low at step2 for Z0 validity |
|
|
226 |
100010101010100 |
2.84765625*S+7.54296875 |
Too low at step2 for Z0 validity |
|
|
227 |
100010101010101 |
2.84765625*S+7.54296875 |
Too low at step2 for Z0 validity |
|
|
228 |
100100000000000 |
0.002197265625*S+0.001708984375 |
Too low at step2 for Z0 validity |
|
|
229 |
100100000000001 |
0.002197265625*S+0.001708984375 |
Too low at step2 for Z0 validity |
|
|
230 |
100100000000010 |
0.01318359375*S+1.01025390625 |
Too low at step2 for Z0 validity |
|
|
231 |
100100000000100 |
0.01318359375*S+0.51025390625 |
Too low at step2 for Z0 validity |
|
|
232 |
100100000000101 |
0.01318359375*S+0.51025390625 |
Too low at step2 for Z0 validity |
|
|
233 |
100100000001000 |
0.01318359375*S+0.26025390625 |
Too low at step2 for Z0 validity |
|
|
234 |
100100000001001 |
0.01318359375*S+0.26025390625 |
Too low at step2 for Z0 validity |
|
|
235 |
100100000001010 |
0.0791015625*S+2.5615234375 |
Too low at step2 for Z0 validity |
|
|
236 |
100100000010000 |
0.01318359375*S+0.13525390625 |
Too low at step2 for Z0 validity |
|
|
237 |
100100000010001 |
0.01318359375*S+0.13525390625 |
Too low at step2 for Z0 validity |
|
|
238 |
100100000010010 |
0.0791015625*S+1.8115234375 |
Too low at step2 for Z0 validity |
|
|
239 |
100100000010100 |
0.0791015625*S+1.3115234375 |
Too low at step2 for Z0 validity |
|
|
240 |
100100000010101 |
0.0791015625*S+1.3115234375 |
Too low at step2 for Z0 validity |
|
|
241 |
100100000100000 |
0.01318359375*S+0.07275390625 |
Too low at step2 for Z0 validity |
|
|
242 |
100100000100001 |
0.01318359375*S+0.07275390625 |
Too low at step2 for Z0 validity |
|
|
243 |
100100000100010 |
0.0791015625*S+1.4365234375 |
Too low at step2 for Z0 validity |
|
|
244 |
100100000100100 |
0.0791015625*S+0.9365234375 |
Too low at step2 for Z0 validity |
|
|
245 |
100100000100101 |
0.0791015625*S+0.9365234375 |
Too low at step2 for Z0 validity |
|
|
246 |
100100000101000 |
0.0791015625*S+0.6865234375 |
Too low at step2 for Z0 validity |
|
|
247 |
100100000101001 |
0.0791015625*S+0.6865234375 |
Too low at step2 for Z0 validity |
|
|
248 |
100100000101010 |
0.474609375*S+5.119140625 |
Too low at step2 for Z0 validity |
|
|
249 |
100100001000000 |
0.01318359375*S+0.04150390625 |
Too low at step2 for Z0 validity |
|
|
250 |
100100001000001 |
0.01318359375*S+0.04150390625 |
Too low at step2 for Z0 validity |
|
|
251 |
100100001000010 |
0.0791015625*S+1.2490234375 |
Too low at step2 for Z0 validity |
|
|
252 |
100100001000100 |
0.0791015625*S+0.7490234375 |
Too low at step2 for Z0 validity |
|
|
253 |
100100001000101 |
0.0791015625*S+0.7490234375 |
Too low at step2 for Z0 validity |
|
|
254 |
100100001001000 |
0.0791015625*S+0.4990234375 |
Too low at step2 for Z0 validity |
|
|
255 |
100100001001001 |
0.0791015625*S+0.4990234375 |
Too low at step2 for Z0 validity |
|
|
256 |
100100001001010 |
0.474609375*S+3.994140625 |
Too low at step2 for Z0 validity |
|
|
257 |
100100001010000 |
0.0791015625*S+0.3740234375 |
Too low at step2 for Z0 validity |
|
|
258 |
100100001010001 |
0.0791015625*S+0.3740234375 |
Too low at step2 for Z0 validity |
|
|
259 |
100100001010100 |
0.474609375*S+2.744140625 |
Too low at step2 for Z0 validity |
|
|
260 |
100100001010101 |
0.474609375*S+2.744140625 |
Too low at step2 for Z0 validity |
|
|
261 |
100100010000000 |
0.01318359375*S+0.02587890625 |
Too low at step2 for Z0 validity |
|
|
262 |
100100010000001 |
0.01318359375*S+0.02587890625 |
Too low at step2 for Z0 validity |
|
|
263 |
100100010000010 |
0.0791015625*S+1.1552734375 |
Too low at step2 for Z0 validity |
|
|
264 |
100100010000100 |
0.0791015625*S+0.6552734375 |
Too low at step2 for Z0 validity |
|
|
265 |
100100010000101 |
0.0791015625*S+0.6552734375 |
Too low at step2 for Z0 validity |
|
|
266 |
100100010001000 |
0.0791015625*S+0.4052734375 |
Too low at step2 for Z0 validity |
|
|
267 |
100100010001001 |
0.0791015625*S+0.4052734375 |
Too low at step2 for Z0 validity |
|
|
268 |
100100010001010 |
0.474609375*S+3.431640625 |
Too low at step2 for Z0 validity |
|
|
269 |
100100010010000 |
0.0791015625*S+0.2802734375 |
Too low at step2 for Z0 validity |
|
|
270 |
100100010010001 |
0.0791015625*S+0.2802734375 |
Too low at step2 for Z0 validity |
|
|
271 |
100100010010010 |
0.474609375*S+2.681640625 |
Too low at step2 for Z0 validity |
|
|
272 |
100100010010100 |
0.474609375*S+2.181640625 |
Too low at step2 for Z0 validity |
|
|
273 |
100100010010101 |
0.474609375*S+2.181640625 |
Too low at step2 for Z0 validity |
|
|
274 |
100100010100000 |
0.0791015625*S+0.2177734375 |
Too low at step2 for Z0 validity |
|
|
275 |
100100010100001 |
0.0791015625*S+0.2177734375 |
Too low at step2 for Z0 validity |
|
|
276 |
100100010100010 |
0.474609375*S+2.306640625 |
Too low at step2 for Z0 validity |
|
|
277 |
100100010100100 |
0.474609375*S+1.806640625 |
Too low at step2 for Z0 validity |
|
|
278 |
100100010100101 |
0.474609375*S+1.806640625 |
Too low at step2 for Z0 validity |
|
|
279 |
100100010101000 |
0.474609375*S+1.556640625 |
Too low at step2 for Z0 validity |
|
|
280 |
100100010101001 |
0.474609375*S+1.556640625 |
Too low at step2 for Z0 validity |
|
|
281 |
100100010101010 |
2.84765625*S+10.33984375 |
Too low at step2 for Z0 validity |
|
|
282 |
100100100000000 |
0.01318359375*S+0.01806640625 |
Too low at step2 for Z0 validity |
|
|
283 |
100100100000001 |
0.01318359375*S+0.01806640625 |
Too low at step2 for Z0 validity |
|
|
284 |
100100100000010 |
0.0791015625*S+1.1083984375 |
Too low at step2 for Z0 validity |
|
|
285 |
100100100000100 |
0.0791015625*S+0.6083984375 |
Too low at step2 for Z0 validity |
|
|
286 |
100100100000101 |
0.0791015625*S+0.6083984375 |
Too low at step2 for Z0 validity |
|
|
287 |
100100100001000 |
0.0791015625*S+0.3583984375 |
Too low at step2 for Z0 validity |
|
|
288 |
100100100001001 |
0.0791015625*S+0.3583984375 |
Too low at step2 for Z0 validity |
|
|
289 |
100100100001010 |
0.474609375*S+3.150390625 |
Too low at step2 for Z0 validity |
|
|
290 |
100100100010000 |
0.0791015625*S+0.2333984375 |
Too low at step2 for Z0 validity |
|
|
291 |
100100100010001 |
0.0791015625*S+0.2333984375 |
Too low at step2 for Z0 validity |
|
|
292 |
100100100010010 |
0.474609375*S+2.400390625 |
Too low at step2 for Z0 validity |
|
|
293 |
100100100010100 |
0.474609375*S+1.900390625 |
Too low at step2 for Z0 validity |
|
|
294 |
100100100010101 |
0.474609375*S+1.900390625 |
Too low at step2 for Z0 validity |
|
|
295 |
100100100100000 |
0.0791015625*S+0.1708984375 |
Too low at step2 for Z0 validity |
|
|
296 |
100100100100001 |
0.0791015625*S+0.1708984375 |
Too low at step2 for Z0 validity |
|
|
297 |
100100100100010 |
0.474609375*S+2.025390625 |
Too low at step2 for Z0 validity |
|
|
298 |
100100100100100 |
0.474609375*S+1.525390625 |
Too low at step2 for Z0 validity |
|
|
299 |
100100100100101 |
0.474609375*S+1.525390625 |
Too low at step2 for Z0 validity |
|
|
300 |
100100100101000 |
0.474609375*S+1.275390625 |
Too low at step2 for Z0 validity |
|
|
301 |
100100100101010 |
2.84765625*S+8.65234375 |
Too low at step2 for Z0 validity |
|
|
302 |
100100101000000 |
0.0791015625*S+0.1396484375 |
Too low at step2 for Z0 validity |
|
|
303 |
100100101000001 |
0.0791015625*S+0.1396484375 |
Too low at step2 for Z0 validity |
|
|
304 |
100100101000010 |
0.474609375*S+1.837890625 |
Too low at step2 for Z0 validity |
|
|
305 |
100100101000100 |
0.474609375*S+1.337890625 |
Too low at step2 for Z0 validity |
|
|
306 |
100100101000101 |
0.474609375*S+1.337890625 |
Too low at step2 for Z0 validity |
|
|
307 |
100100101001000 |
0.474609375*S+1.087890625 |
Too low at step2 for Z0 validity |
|
|
308 |
100100101001001 |
0.474609375*S+1.087890625 |
Too low at step2 for Z0 validity |
|
|
309 |
100100101001010 |
2.84765625*S+7.52734375 |
Too low at step2 for Z0 validity |
|
|
310 |
100100101010000 |
0.474609375*S+0.962890625 |
Too low at step2 for Z0 validity |
|
|
311 |
100100101010001 |
0.474609375*S+0.962890625 |
Too low at step2 for Z0 validity |
|
|
312 |
100100101010100 |
2.84765625*S+6.27734375 |
Too low at step2 for Z0 validity |
|
|
313 |
100100101010101 |
2.84765625*S+6.27734375 |
Too low at step2 for Z0 validity |
|
|
314 |
100101000000000 |
0.01318359375*S+0.01416015625 |
Too low at step2 for Z0 validity |
|
|
315 |
100101000000001 |
0.01318359375*S+0.01416015625 |
Too low at step2 for Z0 validity |
|
|
316 |
100101000000010 |
0.0791015625*S+1.0849609375 |
Too low at step2 for Z0 validity |
|
|
317 |
100101000000100 |
0.0791015625*S+0.5849609375 |
Too low at step2 for Z0 validity |
|
|
318 |
100101000000101 |
0.0791015625*S+0.5849609375 |
Too low at step2 for Z0 validity |
|
|
319 |
100101000001000 |
0.0791015625*S+0.3349609375 |
Too low at step2 for Z0 validity |
|
|
320 |
100101000001001 |
0.0791015625*S+0.3349609375 |
Too low at step2 for Z0 validity |
|
|
321 |
100101000001010 |
0.474609375*S+3.009765625 |
Too low at step2 for Z0 validity |
|
|
322 |
100101000010000 |
0.0791015625*S+0.2099609375 |
Too low at step2 for Z0 validity |
|
|
323 |
100101000010001 |
0.0791015625*S+0.2099609375 |
Too low at step2 for Z0 validity |
|
|
324 |
100101000010010 |
0.474609375*S+2.259765625 |
Too low at step2 for Z0 validity |
|
|
325 |
100101000010100 |
0.474609375*S+1.759765625 |
Too low at step2 for Z0 validity |
|
|
326 |
100101000010101 |
0.474609375*S+1.759765625 |
Too low at step2 for Z0 validity |
|
|
327 |
100101000100000 |
0.0791015625*S+0.1474609375 |
Too low at step2 for Z0 validity |
|
|
328 |
100101000100001 |
0.0791015625*S+0.1474609375 |
Too low at step2 for Z0 validity |
|
|
329 |
100101000100010 |
0.474609375*S+1.884765625 |
Too low at step2 for Z0 validity |
|
|
330 |
100101000100100 |
0.474609375*S+1.384765625 |
Too low at step2 for Z0 validity |
|
|
331 |
100101000101000 |
0.474609375*S+1.134765625 |
Too low at step2 for Z0 validity |
|
|
332 |
100101000101001 |
0.474609375*S+1.134765625 |
Too low at step2 for Z0 validity |
|
|
333 |
100101000101010 |
2.84765625*S+7.80859375 |
Too low at step2 for Z0 validity |
|
|
334 |
100101001000000 |
0.0791015625*S+0.1162109375 |
Too low at step2 for Z0 validity |
|
|
335 |
100101001000001 |
0.0791015625*S+0.1162109375 |
Too low at step2 for Z0 validity |
|
|
336 |
100101001000010 |
0.474609375*S+1.697265625 |
Too low at step2 for Z0 validity |
|
|
337 |
100101001000100 |
0.474609375*S+1.197265625 |
Too low at step2 for Z0 validity |
|
|
338 |
100101001000101 |
0.474609375*S+1.197265625 |
Too low at step2 for Z0 validity |
|
|
339 |
100101001001000 |
0.474609375*S+0.947265625 |
Too low at step2 for Z0 validity |
|
|
340 |
100101001001001 |
0.474609375*S+0.947265625 |
Too low at step2 for Z0 validity |
|
|
341 |
100101001001010 |
2.84765625*S+6.68359375 |
Too low at step2 for Z0 validity |
|
|
342 |
100101001010000 |
0.474609375*S+0.822265625 |
Too low at step2 for Z0 validity |
|
|
343 |
100101001010001 |
0.474609375*S+0.822265625 |
Too low at step2 for Z0 validity |
|
|
344 |
100101001010010 |
2.84765625*S+5.93359375 |
Too low at step2 for Z0 validity |
|
|
345 |
100101001010100 |
2.84765625*S+5.43359375 |
Too low at step2 for Z0 validity |
|
|
346 |
100101001010101 |
2.84765625*S+5.43359375 |
Too low at step2 for Z0 validity |
|
|
347 |
100101010000000 |
0.0791015625*S+0.1005859375 |
Too low at step2 for Z0 validity |
|
|
348 |
100101010000001 |
0.0791015625*S+0.1005859375 |
Too low at step2 for Z0 validity |
|
|
349 |
100101010000010 |
0.474609375*S+1.603515625 |
Too low at step2 for Z0 validity |
|
|
350 |
100101010000100 |
0.474609375*S+1.103515625 |
Too low at step2 for Z0 validity |
|
|
351 |
100101010000101 |
0.474609375*S+1.103515625 |
Too low at step2 for Z0 validity |
|
|
352 |
100101010001000 |
0.474609375*S+0.853515625 |
Too low at step2 for Z0 validity |
|
|
353 |
100101010001001 |
0.474609375*S+0.853515625 |
Too low at step2 for Z0 validity |
|
|
354 |
100101010001010 |
2.84765625*S+6.12109375 |
Too low at step2 for Z0 validity |
|
|
355 |
100101010010000 |
0.474609375*S+0.728515625 |
Too low at step2 for Z0 validity |
|
|
356 |
100101010010001 |
0.474609375*S+0.728515625 |
Too low at step2 for Z0 validity |
|
|
357 |
100101010010010 |
2.84765625*S+5.37109375 |
Too low at step2 for Z0 validity |
|
|
358 |
100101010010100 |
2.84765625*S+4.87109375 |
Too low at step2 for Z0 validity |
|
|
359 |
100101010010101 |
2.84765625*S+4.87109375 |
Too low at step2 for Z0 validity |
|
|
360 |
100101010100000 |
0.474609375*S+0.666015625 |
Too low at step2 for Z0 validity |
|
|
361 |
100101010100001 |
0.474609375*S+0.666015625 |
Too low at step2 for Z0 validity |
|
|
362 |
100101010100010 |
2.84765625*S+4.99609375 |
Too low at step2 for Z0 validity |
|
|
363 |
100101010100100 |
2.84765625*S+4.49609375 |
Too low at step2 for Z0 validity |
|
|
364 |
100101010100101 |
2.84765625*S+4.49609375 |
Too low at step2 for Z0 validity |
|
|
365 |
100101010101000 |
2.84765625*S+4.24609375 |
Too low at step2 for Z0 validity |
|
|
366 |
100101010101010 |
17.0859375*S+26.4765625 |
Too low at step2 for Z0 validity |
|
|
367 |
101000000000000 |
0.002197265625*S+0.001220703125 |
Too low at step5 for Z0 validity |
|
|
368 |
101000000000001 |
0.002197265625*S+0.001220703125 |
Too low at step5 for Z0 validity |
|
|
369 |
101000000000010 |
0.01318359375*S+1.00732421875 |
Too low at step5 for Z0 validity |
|
|
370 |
101000000000100 |
0.01318359375*S+0.50732421875 |
Too low at step5 for Z0 validity |
|
|
371 |
101000000000101 |
0.01318359375*S+0.50732421875 |
Too low at step5 for Z0 validity |
|
|
372 |
101000000001000 |
0.01318359375*S+0.25732421875 |
Too low at step5 for Z0 validity |
|
|
373 |
101000000001001 |
0.01318359375*S+0.25732421875 |
Too low at step5 for Z0 validity |
|
|
374 |
101000000001010 |
0.0791015625*S+2.5439453125 |
Too low at step5 for Z0 validity |
|
|
375 |
101000000010000 |
0.01318359375*S+0.13232421875 |
Too low at step5 for Z0 validity |
|
|
376 |
101000000010001 |
0.01318359375*S+0.13232421875 |
Too low at step5 for Z0 validity |
|
|
377 |
101000000010010 |
0.0791015625*S+1.7939453125 |
Too low at step5 for Z0 validity |
|
|
378 |
101000000010100 |
0.0791015625*S+1.2939453125 |
Too low at step5 for Z0 validity |
|
|
379 |
101000000010101 |
0.0791015625*S+1.2939453125 |
Too low at step5 for Z0 validity |
|
|
380 |
101000000100000 |
0.01318359375*S+0.06982421875 |
Too low at step5 for Z0 validity |
|
|
381 |
101000000100001 |
0.01318359375*S+0.06982421875 |
Too low at step5 for Z0 validity |
|
|
382 |
101000000100010 |
0.0791015625*S+1.4189453125 |
Too low at step5 for Z0 validity |
|
|
383 |
101000000100100 |
0.0791015625*S+0.9189453125 |
Too low at step5 for Z0 validity |
|
|
384 |
101000000100101 |
0.0791015625*S+0.9189453125 |
Too low at step5 for Z0 validity |
|
|
385 |
101000000101000 |
0.0791015625*S+0.6689453125 |
Too low at step5 for Z0 validity |
|
|
386 |
101000000101001 |
0.0791015625*S+0.6689453125 |
Too low at step5 for Z0 validity |
|
|
387 |
101000000101010 |
0.474609375*S+5.013671875 |
Too low at step5 for Z0 validity |
|
|
388 |
101000001000000 |
0.01318359375*S+0.03857421875 |
Too low at step5 for Z0 validity |
|
|
389 |
101000001000001 |
0.01318359375*S+0.03857421875 |
Too low at step5 for Z0 validity |
|
|
390 |
101000001000010 |
0.0791015625*S+1.2314453125 |
Too low at step5 for Z0 validity |
|
|
391 |
101000001000100 |
0.0791015625*S+0.7314453125 |
Too low at step5 for Z0 validity |
|
|
392 |
101000001000101 |
0.0791015625*S+0.7314453125 |
Too low at step5 for Z0 validity |
|
|
393 |
101000001001000 |
0.0791015625*S+0.4814453125 |
Too low at step5 for Z0 validity |
|
|
394 |
101000001001001 |
0.0791015625*S+0.4814453125 |
Too low at step5 for Z0 validity |
|
|
395 |
101000001001010 |
0.474609375*S+3.888671875 |
Too low at step5 for Z0 validity |
|
|
396 |
101000001010000 |
0.0791015625*S+0.3564453125 |
Too low at step5 for Z0 validity |
|
|
397 |
101000001010001 |
0.0791015625*S+0.3564453125 |
Too low at step5 for Z0 validity |
|
|
398 |
101000001010010 |
0.474609375*S+3.138671875 |
Too low at step5 for Z0 validity |
|
|
399 |
101000001010100 |
0.474609375*S+2.638671875 |
Too low at step5 for Z0 validity |
|
|
400 |
101000001010101 |
0.474609375*S+2.638671875 |
Too low at step5 for Z0 validity |
|
|
401 |
101000010000000 |
0.01318359375*S+0.02294921875 |
Too low at step5 for Z0 validity |
|
|
402 |
101000010000001 |
0.01318359375*S+0.02294921875 |
Too low at step5 for Z0 validity |
|
|
403 |
101000010000010 |
0.0791015625*S+1.1376953125 |
Too low at step5 for Z0 validity |
|
|
404 |
101000010000100 |
0.0791015625*S+0.6376953125 |
Too low at step5 for Z0 validity |
|
|
405 |
101000010000101 |
0.0791015625*S+0.6376953125 |
Too low at step5 for Z0 validity |
|
|
406 |
101000010001000 |
0.0791015625*S+0.3876953125 |
Too low at step5 for Z0 validity |
|
|
407 |
101000010001001 |
0.0791015625*S+0.3876953125 |
Too low at step5 for Z0 validity |
|
|
408 |
101000010001010 |
0.474609375*S+3.326171875 |
Too low at step5 for Z0 validity |
|
|
409 |
101000010010000 |
0.0791015625*S+0.2626953125 |
Too low at step5 for Z0 validity |
|
|
410 |
101000010010001 |
0.0791015625*S+0.2626953125 |
Too low at step5 for Z0 validity |
|
|
411 |
101000010010010 |
0.474609375*S+2.576171875 |
Too low at step5 for Z0 validity |
|
|
412 |
101000010010010 |
0.474609375*S+2.576171875 |
Too low at step5 for Z0 validity |
|
|
413 |
101000010010100 |
0.474609375*S+2.076171875 |
Too low at step5 for Z0 validity |
|
|
414 |
101000010010101 |
0.474609375*S+2.076171875 |
Too low at step5 for Z0 validity |
|
|
415 |
101000010100000 |
0.0791015625*S+0.2001953125 |
Too low at step5 for Z0 validity |
|
|
416 |
101000010100001 |
0.0791015625*S+0.2001953125 |
Too low at step5 for Z0 validity |
|
|
417 |
101000010100010 |
0.474609375*S+2.201171875 |
Too low at step5 for Z0 validity |
|
|
418 |
101000010100100 |
0.474609375*S+1.701171875 |
Too low at step5 for Z0 validity |
|
|
419 |
101000010100101 |
0.474609375*S+1.701171875 |
Too low at step5 for Z0 validity |
|
|
420 |
101000010101000 |
0.474609375*S+1.451171875 |
Too low at step5 for Z0 validity |
|
|
421 |
101000010101010 |
2.84765625*S+9.70703125 |
Too low at step5 for Z0 validity |
|
|
422 |
101000100000000 |
0.01318359375*S+0.01513671875 |
Too low at step5 for Z0 validity |
|
|
423 |
101000100000001 |
0.01318359375*S+0.01513671875 |
Too low at step5 for Z0 validity |
|
|
424 |
101000100000010 |
0.0791015625*S+1.0908203125 |
Too low at step5 for Z0 validity |
|
|
425 |
101000100000100 |
0.0791015625*S+0.5908203125 |
Too low at step5 for Z0 validity |
|
|
426 |
101000100000101 |
0.0791015625*S+0.5908203125 |
Too low at step5 for Z0 validity |
|
|
427 |
101000100001000 |
0.0791015625*S+0.3408203125 |
Too low at step5 for Z0 validity |
|
|
428 |
101000100001001 |
0.0791015625*S+0.3408203125 |
Too low at step5 for Z0 validity |
|
|
429 |
101000100001010 |
0.474609375*S+3.044921875 |
Too low at step5 for Z0 validity |
|
|
430 |
101000100010000 |
0.0791015625*S+0.2158203125 |
Too low at step5 for Z0 validity |
|
|
431 |
101000100010001 |
0.0791015625*S+0.2158203125 |
Too low at step5 for Z0 validity |
|
|
432 |
101000100010010 |
0.474609375*S+2.294921875 |
Too low at step5 for Z0 validity |
|
|
433 |
101000100010100 |
0.474609375*S+1.794921875 |
Too low at step5 for Z0 validity |
|
|
434 |
101000100010101 |
0.474609375*S+1.794921875 |
Too low at step5 for Z0 validity |
|
|
435 |
101000100100000 |
0.0791015625*S+0.1533203125 |
Too low at step5 for Z0 validity |
|
|
436 |
101000100100001 |
0.0791015625*S+0.1533203125 |
Too low at step5 for Z0 validity |
|
|
437 |
101000100100010 |
0.474609375*S+1.919921875 |
Too low at step5 for Z0 validity |
|
|
438 |
101000100100100 |
0.474609375*S+1.419921875 |
Too low at step5 for Z0 validity |
|
|
439 |
101000100100101 |
0.474609375*S+1.419921875 |
Too low at step5 for Z0 validity |
|
|
440 |
101000100101000 |
0.474609375*S+1.169921875 |
Too low at step5 for Z0 validity |
|
|
441 |
101000100101010 |
2.84765625*S+8.01953125 |
Too low at step5 for Z0 validity |
|
|
442 |
101001000000000 |
0.01318359375*S+0.01123046875 |
Too low at step7 for Z0 validity |
|
|
443 |
101001000000001 |
0.01318359375*S+0.01123046875 |
Too low at step7 for Z0 validity |
|
|
444 |
101001000000010 |
0.0791015625*S+1.0673828125 |
Too low at step7 for Z0 validity |
|
|
445 |
101001000000100 |
0.0791015625*S+0.5673828125 |
Too low at step7 for Z0 validity |
|
|
446 |
101001000000101 |
0.0791015625*S+0.5673828125 |
Too low at step7 for Z0 validity |
|
|
447 |
101001000001000 |
0.0791015625*S+0.3173828125 |
Too low at step7 for Z0 validity |
|
|
448 |
101001000001001 |
0.0791015625*S+0.3173828125 |
Too low at step7 for Z0 validity |
|
|
449 |
101001000001010 |
0.474609375*S+2.904296875 |
Too low at step7 for Z0 validity |
|
|
450 |
101001000010000 |
0.0791015625*S+0.1923828125 |
Too low at step7 for Z0 validity |
|
|
451 |
101001000010001 |
0.0791015625*S+0.1923828125 |
Too low at step7 for Z0 validity |
|
|
452 |
101001000010010 |
0.474609375*S+2.154296875 |
Too low at step7 for Z0 validity |
|
|
453 |
101001000010100 |
0.474609375*S+1.654296875 |
Too low at step7 for Z0 validity |
|
|
454 |
101001000010101 |
0.474609375*S+1.654296875 |
Too low at step7 for Z0 validity |
|
|
455 |
101001000100000 |
0.0791015625*S+0.1298828125 |
Too low at step7 for Z0 validity |
|
|
456 |
101001000100001 |
0.0791015625*S+0.1298828125 |
Too low at step7 for Z0 validity |
|
|
457 |
101001000100010 |
0.474609375*S+1.779296875 |
Too low at step7 for Z0 validity |
|
|
458 |
101001000100100 |
0.474609375*S+1.279296875 |
Too low at step7 for Z0 validity |
|
|
459 |
101001000100101 |
0.474609375*S+1.279296875 |
Too low at step7 for Z0 validity |
|
|
460 |
101001000101000 |
0.474609375*S+1.029296875 |
Too low at step7 for Z0 validity |
|
|
461 |
101001000101001 |
0.474609375*S+1.029296875 |
Too low at step7 for Z0 validity |
|
|
462 |
101001000101010 |
2.84765625*S+7.17578125 |
Too low at step7 for Z0 validity |
|
|
463 |
101001001000000 |
0.0791015625*S+0.0986328125 |
Too low at step7 for Z0 validity |
|
|
464 |
101001001000001 |
0.0791015625*S+0.0986328125 |
Too low at step7 for Z0 validity |
|
|
465 |
101001001000010 |
0.474609375*S+1.591796875 |
Too low at step7 for Z0 validity |
|
|
466 |
101001001000100 |
0.474609375*S+1.091796875 |
Too low at step7 for Z0 validity |
|
|
467 |
101001001000101 |
0.474609375*S+1.091796875 |
Too low at step7 for Z0 validity |
|
|
468 |
101001001001000 |
0.474609375*S+0.841796875 |
Too low at step7 for Z0 validity |
|
|
469 |
101001001001001 |
0.474609375*S+0.841796875 |
Too low at step7 for Z0 validity |
|
|
470 |
101001001001010 |
2.84765625*S+6.05078125 |
Too low at step7 for Z0 validity |
|
|
471 |
101001001010000 |
0.474609375*S+0.716796875 |
Too low at step7 for Z0 validity |
|
|
472 |
101001001010001 |
0.474609375*S+0.716796875 |
Too low at step7 for Z0 validity |
|
|
473 |
101001001010010 |
2.84765625*S+5.30078125 |
Too low at step7 for Z0 validity |
|
|
474 |
101001001010100 |
2.84765625*S+4.80078125 |
Too low at step7 for Z0 validity |
|
|
475 |
101001001010101 |
2.84765625*S+4.80078125 |
Too low at step7 for Z0 validity |
|
|
476 |
101001010000000 |
0.0791015625*S+0.0830078125 |
Too low at step9 for Z0 validity |
|
|
477 |
101001010000001 |
0.0791015625*S+0.0830078125 |
Too low at step9 for Z0 validity |
|
|
478 |
101001010000010 |
0.474609375*S+1.498046875 |
Too low at step9 for Z0 validity |
|
|
479 |
101001010000100 |
0.474609375*S+0.998046875 |
Too low at step9 for Z0 validity |
|
|
480 |
101001010000101 |
0.474609375*S+0.998046875 |
Too low at step9 for Z0 validity |
|
|
481 |
101001010001000 |
0.474609375*S+0.748046875 |
Too low at step9 for Z0 validity |
|
|
482 |
101001010001001 |
0.474609375*S+0.748046875 |
Too low at step9 for Z0 validity |
|
|
483 |
101001010001010 |
2.84765625*S+5.48828125 |
Too low at step9 for Z0 validity |
|
|
484 |
101001010010000 |
0.474609375*S+0.623046875 |
Too low at step12 for Z0 validity |
|
|
485 |
101001010010001 |
0.474609375*S+0.623046875 |
Too low at step12 for Z0 validity |
|
|
486 |
101001010010010 |
2.84765625*S+4.73828125 |
Too low at step12 for Z0 validity |
|
|
487 |
101001010010100 |
2.84765625*S+4.23828125 |
Passes Lemma 5a |
|
|
488 |
101001010010101 |
2.84765625*S+4.23828125 |
Passes Lemma 5a |
|
|
489 |
101001010100000 |
0.474609375*S+0.560546875 |
Too low at step12 for Z0 validity |
|
|
490 |
101001010100001 |
0.474609375*S+0.560546875 |
Too low at step12 for Z0 validity |
|
|
491 |
101001010100010 |
2.84765625*S+4.36328125 |
Too low at step12 for Z0 validity |
|
|
492 |
101001010100100 |
2.84765625*S+3.86328125 |
Passes Lemma 5a |
|
|
493 |
101001010100101 |
2.84765625*S+3.86328125 |
Passes Lemma 5a |
|
|
494 |
101001010101000 |
2.84765625*S+3.61328125 |
Passes Lemma 5a |
|
|
495 |
101001010101001 |
2.84765625*S+3.61328125 |
Passes Lemma 5a |
|
|
496 |
101001010101010 |
17.0859375*S+22.6796875 |
Passes Lemma 5a |
|
|
497 |
101010000000000 |
0.01318359375*S+0.00927734375 |
Too low at step7 for Z0 validity |
|
|
498 |
101010000000001 |
0.01318359375*S+0.00927734375 |
Too low at step7 for Z0 validity |
|
|
499 |
101010000000010 |
0.0791015625*S+1.0556640625 |
Too low at step7 for Z0 validity |
|
|
500 |
101010000000100 |
0.0791015625*S+0.5556640625 |
Too low at step7 for Z0 validity |
|
|
501 |
101010000000101 |
0.0791015625*S+0.5556640625 |
Too low at step7 for Z0 validity |
|
|
502 |
101010000001000 |
0.0791015625*S+0.3056640625 |
Too low at step7 for Z0 validity |
|
|
503 |
101010000001001 |
0.0791015625*S+0.3056640625 |
Too low at step7 for Z0 validity |
|
|
504 |
101010000001010 |
0.474609375*S+2.833984375 |
Too low at step7 for Z0 validity |
|
|
505 |
101010000010000 |
0.0791015625*S+0.1806640625 |
Too low at step7 for Z0 validity |
|
|
506 |
101010000010001 |
0.0791015625*S+0.1806640625 |
Too low at step7 for Z0 validity |
|
|
507 |
101010000010010 |
0.474609375*S+2.083984375 |
Too low at step7 for Z0 validity |
|
|
508 |
101010000010100 |
0.474609375*S+1.583984375 |
Too low at step7 for Z0 validity |
|
|
509 |
101010000010101 |
0.474609375*S+1.583984375 |
Too low at step7 for Z0 validity |
|
|
510 |
101010000100000 |
0.0791015625*S+0.1181640625 |
Too low at step7 for Z0 validity |
|
|
511 |
101010000100001 |
0.0791015625*S+0.1181640625 |
Too low at step7 for Z0 validity |
|
|
512 |
101010000100010 |
0.474609375*S+1.708984375 |
Too low at step7 for Z0 validity |
|
|
513 |
101010000100100 |
0.474609375*S+1.208984375 |
Too low at step7 for Z0 validity |
|
|
514 |
101010000100101 |
0.474609375*S+1.208984375 |
Too low at step7 for Z0 validity |
|
|
515 |
101010000101000 |
0.474609375*S+0.958984375 |
Too low at step7 for Z0 validity |
|
|
516 |
101010000101001 |
0.474609375*S+0.958984375 |
Too low at step7 for Z0 validity |
|
|
517 |
101010000101010 |
2.84765625*S+6.75390625 |
Too low at step7 for Z0 validity |
|
|
518 |
101010001000000 |
0.0791015625*S+0.0869140625 |
Too low at step7 for Z0 validity |
|
|
519 |
101010001000001 |
0.0791015625*S+0.0869140625 |
Too low at step7 for Z0 validity |
|
|
520 |
101010001000010 |
0.474609375*S+1.521484375 |
Too low at step7 for Z0 validity |
|
|
521 |
101010001000100 |
0.474609375*S+1.021484375 |
Too low at step7 for Z0 validity |
|
|
522 |
101010001000101 |
0.474609375*S+1.021484375 |
Too low at step7 for Z0 validity |
|
|
523 |
101010001001000 |
0.474609375*S+0.771484375 |
Too low at step7 for Z0 validity |
|
|
524 |
101010001001010 |
2.84765625*S+5.62890625 |
Too low at step7 for Z0 validity |
|
|
525 |
101010001010000 |
0.474609375*S+0.646484375 |
Too low at step7 for Z0 validity |
|
|
526 |
101010001010001 |
0.474609375*S+0.646484375 |
Too low at step7 for Z0 validity |
|
|
527 |
101010001010010 |
2.84765625*S+4.87890625 |
Too low at step7 for Z0 validity |
|
|
528 |
101010001010100 |
2.84765625*S+4.37890625 |
Too low at step7 for Z0 validity |
|
|
529 |
101010001010101 |
2.84765625*S+4.37890625 |
Too low at step7 for Z0 validity |
|
|
530 |
101010010000000 |
0.0791015625*S+0.0712890625 |
Too low at step10 for Z0 validity |
|
|
531 |
101010010000001 |
0.0791015625*S+0.0712890625 |
Too low at step10 for Z0 validity |
|
|
532 |
101010010000010 |
0.474609375*S+1.427734375 |
Too low at step10 for Z0 validity |
|
|
533 |
101010010000100 |
0.474609375*S+0.927734375 |
Too low at step10 for Z0 validity |
|
|
534 |
101010010001000 |
0.474609375*S+0.677734375 |
Too low at step10 for Z0 validity |
|
|
535 |
101010010001001 |
0.474609375*S+0.677734375 |
Too low at step10 for Z0 validity |
|
|
536 |
101010010001010 |
2.84765625*S+5.06640625 |
Too low at step10 for Z0 validity |
|
|
537 |
101010010010000 |
0.474609375*S+0.552734375 |
Too low at step12 for Z0 validity |
|
|
538 |
101010010010001 |
0.474609375*S+0.552734375 |
Too low at step12 for Z0 validity |
|
|
539 |
101010010010010 |
2.84765625*S+4.31640625 |
Too low at step12 for Z0 validity |
|
|
540 |
101010010010100 |
2.84765625*S+3.81640625 |
Passes Lemma 5a |
|
|
541 |
101010010010101 |
2.84765625*S+3.81640625 |
Passes Lemma 5a |
|
|
542 |
101010010100000 |
0.474609375*S+0.490234375 |
Too low at step12 for Z0 validity |
|
|
543 |
101010010100001 |
0.474609375*S+0.490234375 |
Too low at step12 for Z0 validity |
|
|
544 |
101010010100010 |
2.84765625*S+3.94140625 |
Too low at step12 for Z0 validity |
|
|
545 |
101010010100100 |
2.84765625*S+3.44140625 |
Passes Lemma 5a |
|
|
546 |
101010010100101 |
2.84765625*S+3.44140625 |
Passes Lemma 5a |
|
|
547 |
101010010101000 |
2.84765625*S+3.19140625 |
Passes Lemma 5a |
|
|
548 |
101010010101001 |
2.84765625*S+3.19140625 |
Passes Lemma 5a |
|
|
549 |
101010010101010 |
17.0859375*S+20.1484375 |
Passes Lemma 5a |
|
|
550 |
101010100000000 |
0.0791015625*S+0.0634765625 |
Too low at step10 for Z0 validity |
|
|
551 |
101010100000001 |
0.0791015625*S+0.0634765625 |
Too low at step10 for Z0 validity |
|
|
552 |
101010100000010 |
0.474609375*S+1.380859375 |
Too low at step10 for Z0 validity |
|
|
553 |
101010100000100 |
0.474609375*S+0.880859375 |
Too low at step10 for Z0 validity |
|
|
554 |
101010100000101 |
0.474609375*S+0.880859375 |
Too low at step10 for Z0 validity |
|
|
555 |
101010100001000 |
0.474609375*S+0.630859375 |
Too low at step10 for Z0 validity |
|
|
556 |
101010100001001 |
0.474609375*S+0.630859375 |
Too low at step10 for Z0 validity |
|
|
557 |
101010100001010 |
2.84765625*S+4.78515625 |
Too low at step10 for Z0 validity |
|
|
558 |
101010100010000 |
0.474609375*S+0.505859375 |
Too low at step12 for Z0 validity |
|
|
559 |
101010100010001 |
0.474609375*S+0.505859375 |
Too low at step12 for Z0 validity |
|
|
560 |
101010100010010 |
2.84765625*S+4.03515625 |
Too low at step12 for Z0 validity |
|
|
561 |
101010100010100 |
2.84765625*S+3.53515625 |
Passes Lemma 5a |
|
|
562 |
101010100010101 |
2.84765625*S+3.53515625 |
Passes Lemma 5a |
|
|
563 |
101010100100000 |
0.474609375*S+0.443359375 |
Too low at step12 for Z0 validity |
|
|
564 |
101010100100001 |
0.474609375*S+0.443359375 |
Too low at step12 for Z0 validity |
|
|
565 |
101010100100010 |
2.84765625*S+3.66015625 |
Too low at step12 for Z0 validity |
|
|
566 |
101010100100100 |
2.84765625*S+3.16015625 |
Passes Lemma 5a |
|
|
567 |
101010100100101 |
2.84765625*S+3.16015625 |
Passes Lemma 5a |
|
|
568 |
101010100101000 |
2.84765625*S+2.91015625 |
Passes Lemma 5a |
|
|
569 |
101010100101001 |
2.84765625*S+2.91015625 |
Passes Lemma 5a |
|
|
570 |
101010100101010 |
17.0859375*S+18.4609375 |
Passes Lemma 5a |
|
|
571 |
101010101000000 |
0.474609375*S+0.412109375 |
Too low at step12 for Z0 validity |
|
|
572 |
101010101000001 |
0.474609375*S+0.412109375 |
Too low at step12 for Z0 validity |
|
|
573 |
101010101000010 |
2.84765625*S+3.47265625 |
Too low at step12 for Z0 validity |
|
|
574 |
101010101000100 |
2.84765625*S+2.97265625 |
All remaining paths |
|
|
575 |
101010101000101 |
2.84765625*S+2.97265625 |
pass Lemma 5a |
|
|
576 |
101010101001000 |
2.84765625*S+2.72265625 |
||
|
577 |
101010101001001 |
2.84765625*S+2.72265625 |
||
|
578 |
101010101001010 |
17.0859375*S+17.3359375 |
||
|
579 |
101010101010000 |
2.84765625*S+2.59765625 |
||
|
580 |
101010101010001 |
2.84765625*S+2.59765625 |
||
|
581 |
101010101010100 |
17.0859375*S+16.0859375 |
||
|
582 |
101010101010101 |
17.0859375*S+16.0859375 |
||
Lemma 7 – If Z(0) exists its
first 14 transforms must follow one of these 30 pathways.
This is a restatement of the
above table, showing only the exceptions to Lemma 5a – for clarity. C-numbers which follow the
Transform patterns shown up to
step 14 have been added for ease of verification of values. Possible Z(0) pathways have been
Numbered for testing against
Lemma 5b and 5c constraints.
|
Z-PATH1
like 507 |
101001010010100 |
2.84765625*S+4.23828125 |
|
Z-PATH2
like 251 |
101001010010101 |
2.84765625*S+4.23828125 |
|
Z-PATH3
like 347 |
101001010100100 |
2.84765625*S+3.86328125 |
|
Z-PATH4
like 91 |
101001010100101 |
2.84765625*S+3.86328125 |
|
Z-PATH5
like 411 |
101001010101000 |
2.84765625*S+3.61328125 |
|
Z-PATH6
like 155 |
101001010101001 |
2.84765625*S+3.61328125 |
|
Z-PATH7
like 27 |
101001010101010 |
17.0859375*S+22.6796875 |
|
Z-PATH8
like 71 |
101010010010100 |
2.84765625*S+3.81640625 |
|
Z-PATH9
like 327 |
101010010010101 |
2.84765625*S+3.81640625 |
|
Z-PATH10
like 423 |
101010010100100 |
2.84765625*S+3.44140625 |
|
Z-PATH11
like 167 |
101010010100101 |
2.84765625*S+3.44140625 |
|
Z-PATH12
like 487 |
101010010101000 |
2.84765625*S+3.19140625 |
|
Z-PATH13
like 231 |
101010010101001 |
2.84765625*S+3.19140625 |
|
Z-PATH14
like 103 |
101010010101010 |
17.0859375*S+20.1484375 |
|
Z-PATH15
like 463 |
101010100010100 |
2.84765625*S+3.53515625 |
|
Z-PATH16
like 207 |
101010100010101 |
2.84765625*S+3.53515625 |
|
Z-PATH17
like 303 |
101010100100100 |
2.84765625*S+3.16015625 |
|
Z-PATH18
like 47 |
101010100100101 |
2.84765625*S+3.16015625 |
|
Z-PATH19
like 367 |
101010100101000 |
2.84765625*S+2.91015625 |
|
Z-PATH20
like 111 |
101010100101001 |
2.84765625*S+2.91015625 |
|
Z-PATH21
like 239 |
101010100101010 |
17.0859375*S+18.4609375 |
|
Z-PATH22
like 479 |
101010101000100 |
2.84765625*S+2.97265625 |
|
Z-PATH23
like 223 |
101010101000101 |
2.84765625*S+2.97265625 |
|
Z-PATH24
like 287 |
101010101001000 |
2.84765625*S+2.72265625 |
|
Z-PATH25
like 31 |
101010101001001 |
2.84765625*S+2.72265625 |
|
Z-PATH26
like 159 |
101010101001010 |
17.0859375*S+17.3359375 |
|
Z-PATH27
like 63 |
101010101010000 |
2.84765625*S+2.59765625 |
|
Z-PATH28
like 319 |
101010101010001 |
2.84765625*S+2.59765625 |
|
Z-PATH29
like 127 |
101010101010100 |
17.0859375*S+16.0859375 |
|
Z-PATH30
like 255 |
101010101010101 |
17.0859375*S+16.0859375 |
Lemma 8 — The simple extended Z-PATH30
above can be shown to disallow Z0,
For any odd positive integer S,
whose path extends alternating odd and even 101010101010101......n. it would
appear that the path must be a z-number path, since it must increase, at least
to n.
However where n > 145,
S-1 will follow the path 01010101010101..................................n,
and at step 145 both paths join.
If S = Z0 , S-1 =
a c-number, hence a contradiction exists, as step145 must be both a z-number
and a c-number.
It should be stated that Lemma 7
does not show that if a different path permits the existence
of a lowest z-number, that a case in which S and S-1 are
both z-numbers greater than Z0 is impossible, only that Case 582 cannot permit Z0 and only where extension to n >145, permits
parallel trajectories of S and S-1 to step 145.
Lemma — Contradiction
Let a lowest z-number Z(0) exist.
It must firstly obey the boundaries shown in Lemma 5a, that is, as developed by Lemma 6 and 7 its the first 14 steps can only follow
30 possible paths. By Lemma 8 it cannot
follow an extended case of the 30th path.
If Z0 must
therefore follow one of the 30 paths shown in Lemma 8, however the paths must
also be shown avoid Lemma 5b and 5c constraints, including not joining with trajectories
from surrounding c-numbers.
Initial investigation shows that
where the 30 trajectory paths are followed by illustrative c-numbers they would in each case coalesce with trajectories originating in ranges which would be
required to be c-numbers if positioned parallel to Z(0). Such coalescence
is invalid for z-numbers arising from Z(0) as a number cannot both be
produced by a c-number and a z-number.
If no path is possible for Z(0) which does not either a) produce an even z-number
<2Z(0) invalidation of Lemma 2 (Expanded Lemma 5a,6) or b) coalesces with a
c-number path necessitated by Lemma 5b and 5c, 6) then no least z-number Z(0)
exists
If no least z-number exists, no
z-number exists.
If no z-number exists, then
Collatz is proven.
Conclusion
Strong form
Since assuming the existence of
a least z‑number leads to contradiction, no such number exists. Thus no z‑numbers
exist. Therefore every positive integer is a c‑number, and the Collatz
conjecture is true.
Weak form
No Z(0) can exist without contradiction unless it both a)
follows one of 30 possible patterns of 14 step odd-even transforms and b)
avoids joining trajectories of i) every even number between Z(0)+2,2Z(0)-2 and
their multiples, ii) the even multiples (precursors) of every odd number <
Z(0), iii) the even multiples (precursors) of every other odd number higher
than Z(0) beginning at 2(Z(0)+2), iv) the onward transforms of every other odd
number higher than Z(0) beginning at Z(0)+2.
Further exploration of these limits will exert significant constraints
on Z(0).
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