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The b-numbers below 256

The numbers k with k == 5 (mod 8), and s(k-2) = k with s(.) the hypothetical sequence generated by Turing Machine #4, are what I’ve called “b-numbers” for basic numbers.

I haven’t succeeded in finding any rule that determines the whole sequence. Below, I copy the 26 b-numbers below 256. I’ve found this to be a hard problem.

b-numbers below 256:

5
13
21
29
37
45
53
61
77
85
93
101
109
117
125
149
173
181
189
205
213
221
229
237
245
253

 

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Written by meditationatae

September 27, 2017 at 11:35 pm

Posted in History

b-numbers with up to 10 bits

I refer to an earlier post a few days ago as an introduction to the problem:

https://meditationatae.wordpress.com/2017/09/23/the-b-numbers-again/

b-numbers are positive integers k with k == 5 (mod 8) such that  s(k-2) = k; here, s(1), s(2), s(3), … is the hypothetical integer sequence computed by TM #4 (chaotic) of Heiner Marxen and Buntrock (discovered around 1990).

 

There are 42 b-numbers in the interval [1, 511], that is b-numbers with 1 through 9 bits. There are 22 b-numbers in the interval [512, 1023], that is numbers with 10 bits.

If y is a 10-bit b-number, then y > 512 in fact, and y-512 is a number in the range [1, 511] with 1 to 9 bits.

It turns out that y-512 is a b-number with 1 to 9 bits, when y is a 10-bit b-number. There are 22 possible 10-bit

b-numbers, and each of them gives us a different 1 to 9-bit b-number when we subtract 512 from it.

There are 42 b-numbers in the interval [1, 511], so if x is one of them, x+512 is a 10-bit number satisfying x == 5 (mod 8). For 22 of those 42, x+512 is also a b-number, and in the remaining 20 cases, x+512 isn’t a b-number.

For the 42 values of x, I’m trying to figure out when

x and x+512 are both b-numbers, contrasted with x being a b-number and x+512 not being a b-number.

The table below has 42 rows, one per b-number in the interval [1,511]. First comes the binary expansion of

x+512, x being a b-number with 1 to 9 bits. Then comes x in decimal, x being a 1 to 9-bit b-number. Then comes x+512, a 10-bit number. Next comes ‘1’ if x+512 is not a b-number, and ‘2’ if x+512 is indeed a 10-bit b-number, which happens in 22 cases.

I looked for a relationship between the ‘2’/’1′ which tells us whether x+512 is a b-number, and x+512 ‘s binary expansion.

Up to now, for this and the 13-bit case, I’ve found no

logical relationship.

The table with 42 rows is copied below:

 

1 0 0 0 0 0 0 1 0 1 5 517 1
1 0 0 0 0 0 1 1 0 1 13 525 1
1 0 0 0 0 1 0 1 0 1 21 533 1
1 0 0 0 0 1 1 1 0 1 29 541 1
1 0 0 0 1 0 0 1 0 1 37 549 1
1 0 0 0 1 0 1 1 0 1 45 557 1
1 0 0 0 1 1 0 1 0 1 53 565 1
1 0 0 0 1 1 1 1 0 1 61 573 1
1 0 0 1 0 0 1 1 0 1 77 589 1
1 0 0 1 0 1 0 1 0 1 85 597 1
1 0 0 1 0 1 1 1 0 1 93 605 1
1 0 0 1 1 0 0 1 0 1 101 613 1
1 0 0 1 1 0 1 1 0 1 109 621 1
1 0 0 1 1 1 0 1 0 1 117 629 2
1 0 0 1 1 1 1 1 0 1 125 637 1
1 0 1 0 0 1 0 1 0 1 149 661 1
1 0 1 0 1 0 1 1 0 1 173 685 1
1 0 1 0 1 1 0 1 0 1 181 693 2
1 0 1 0 1 1 1 1 0 1 189 701 1
1 0 1 1 0 0 1 1 0 1 205 717 1
1 0 1 1 0 1 0 1 0 1 213 725 2
1 0 1 1 0 1 1 1 0 1 221 733 1
1 0 1 1 1 0 0 1 0 1 229 741 1
1 0 1 1 1 0 1 1 0 1 237 749 2
1 0 1 1 1 1 0 1 0 1 245 757 2
1 0 1 1 1 1 1 1 0 1 253 765 2
1 1 0 0 1 1 0 1 0 1 309 821 2
1 1 0 1 0 1 0 1 0 1 341 853 2
1 1 0 1 1 0 1 1 0 1 365 877 2
1 1 0 1 1 1 0 1 0 1 373 885 2
1 1 0 1 1 1 1 1 0 1 381 893 2
1 1 1 0 0 1 0 1 0 1 405 917 2
1 1 1 0 1 0 1 1 0 1 429 941 2
1 1 1 0 1 1 0 1 0 1 437 949 2
1 1 1 0 1 1 1 1 0 1 445 957 2
1 1 1 1 0 0 1 1 0 1 461 973 2
1 1 1 1 0 1 0 1 0 1 469 981 2
1 1 1 1 0 1 1 1 0 1 477 989 2
1 1 1 1 1 0 0 1 0 1 485 997 2
1 1 1 1 1 0 1 1 0 1 493 1005 2
1 1 1 1 1 1 0 1 0 1 501 1013 2
1 1 1 1 1 1 1 1 0 1 509 1021 2

 

 

Written by meditationatae

September 26, 2017 at 6:02 am

Posted in History

My GPG ciphertext test (on new key)

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lAVBy6ZprrWtB+5PDnflZsA3eRr9M5iRvHaDAVHfloyWRuuAKdu1GWeJeWTT6ffe
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1PYbWdJTdrI7vMrRzuw4i1cASuK30gFlVxshcDidhxkWOIwvgu+ownfQ/tGeliKF
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Written by meditationatae

September 25, 2017 at 9:36 am

Posted in History

cat test2 | sha256sum 4607a2d0c064c3de08b1537bff179b84b54e2a0fc8220e69f6f06a0938fec86d –

Hello World!

Written by meditationatae

September 25, 2017 at 7:14 am

Posted in History

$ cat topost | sha256sum 1e7c2e0e2352235a5f0f537a2c434b3f73307c91923da47762e873c14b187e80 –

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IDI4IDI5IDMwICAgMjkgMzAgMzEKCgo=

Written by meditationatae

September 25, 2017 at 6:47 am

Posted in History

More on b-numbers, part B

This is a continuation of the previous post on b-numbers.

===

For a b-number ‘x’ in the range 4096 to 8191 (13 bit length),
it appears at first glance that, modulo 4096,

x == y (mod 4096), where y is a b-number in
the range 1-4095;

Details:

 

k up to 4095:

 

5 // term 1
13
21
29
37
45
53
61
77
85 // term 10
93
101
109
117
125
149
173
181
189
205
213
221
229
237 x
245
253
309
341
365
373
381
405
429
437 x
445
461
469
477
485
493 x
501
509
629
693
725
749
757
765
821
853 // term 50
877
885
893
917
941
949
957
973
981
989
997
1005
1013
1021
1205
1237
1261
1277
1397
1461
1493
1517
1525
1533
1653
1717
1749
1773
1781
1789
1845
1877
1901
1909
1917
1941
1965
1973
1981
1997
2005
2013
2021
2029
2037
2045
2229
2285
2485 /
2517 // term 100
2541
2557
2741
2773
2797
2813
2933
2997
3029
3053
3061
3069
3253
3285
3309
3325
3445
3509
3541
3565
3573
3581
3701
3765
3797
3821
3829
3837
3893
3925 // t. 130
3949
3957
3965
3989
4013
4021
4029
4045
4053
4061 // term 140
4069
4077
4085
4093 // term 144

 

k from 4096 to 8191, modulo 4096

 

237 // b-number term number 145, modulo 4096
437
493
725
765
885
1013
1205
1261
1461
1493
1517
1533
1717
1749
1773
1789
1909
1973
2005
2029
2037
2045
2229
2285
2485
2517
2541
2557
2741
2773
2797
2813
2933
2997
3029
3053
3061
3069
3253
3285
3309
3325
3445
3509
3541
3565
3573
3581
3701
3765
3797
3821
3829
3837
3893
3925
3949
3957
3965
3989
4013
4021
4029
4045
4053
4061
4069
4077
4085
4093

 

Written by meditationatae

September 25, 2017 at 12:06 am

Posted in History

The b-numbers again

If we write the (hypothetical, from now on understood) sequence from TM #4 (chaotic), 5-state, 2-symbol tape of Marxen and Buntrock in rows of 8, the 3rd column has a complex structure.

Suppose we use “index 1”-notation, which just means that the first term of the sequence, 3, is written s(1) instead of s(0) as would be the case with arrays in C, C++, and so on.

Then the third column is made up of s(3) = 5, s(11), s(19), and in general s(8j + 3) for some integer j with 0<= j < oo.

Empirically,  s(j) <= j+2. When j is congruent to 3 modulo 8, so that s(j) is a third-column term, it happens that:

s(j) = j+2 exactly.

When this happens, in column 3, I call j+2 a b-number (basic number). Or in other words, if k is such that s(k-2) = k, and

k == 5 (mod 3), and of course k >2, then by definition k is a basic number or b-number.

When k == 5 (mod 3), but s(k-2) is not k, then I’ve called k a forbidden number. This is justified by studying the binary expansion of s(k-2)  when k is a forbidden number.

Empirical study gives some credence to the belief that, when k is not a b-number, s(k-2) can be obtained from ‘k’ by omitting one or more of the most significant bits of k, all consecutive bits, while always leaving perhaps 3 bits of ‘k’ to yield 101_(Base 2) = 5_(Base 10).

Moreover, a first look gave me a clue that when k is forbidden, then s(k-2) can be obtained from the binary expansion of k by omitting the most significant ‘1’ bit from k in base 2, and going on striking-out mentally the most significant ‘1’ bits (starting at the leftmost bit) and stopping when and only when the modified number is a b-number. Thus, if k is forbidden, s(k-2) would still be a b-number or basic number.

The new feature I hadn’t looked at previously is the count of b-numbers by bit-length.

The following table has this , with a coming explanation:

n count(n)
======================
10 22
11 32
12 48
13 71
14 106
15 158
16 234
17 348
18 518
19 772
20 1152

n is a bit-length of a positive integer.
Examples: 6 has bit-length 3, 2^k has
bit length k+1, 100 has bit-length 7,
ans 2^k – 1 has bit-length k.

count(n) is the number of b-numbers with bit-length equal to n.
So there are 22 b-numbers with 10 bits, which means in the
range 512 to 1023 inclusive. And so on up to 20 bits.

You’ll see that c(n+1) is approximately (3/2)*c(n),
better than chance in my opinion.

In other words, while there are 48 b-numbers
in the range 2048 to 4095, there are only
23 in the range 4096 to 8191 (which is
twice as long).
To if we increase n by +1, the frequency of
b-numbers is reduced by a factor of 4 approximately.

Perhaps these correspond to two extra “pseudo-random”
boolean 0/1 conditions for a 13-bit b-number, as
compared to a 12-bit b-number. This line of
inquiry has not so far been pursued further.

If you look at the first few dozen b-numbers,
you might notice that 2^k – 3 is a b-number
for k = 3, 4, 5, 6, 7, … 15, …
There are other “magic numbers” like 3.
While consistent, these “magic numbers”
and their patterns still give complicated
sequences.

Perhaps I missed something that could be guessed
from the fact that c(n+1) is approximately (3/2)*c(n).

I find this problem of pattern recognition both
tantalizing and frustrating, because it is
produced by such a seemingly simple 5-state,
2-symbol Turing machine, starting with a
blank tape.

The first 100 b-numbers :

5
13
21
29
37
45
53
61
77
85
93
101
109
117
125
149
173
181
189
205
213
221
229
237
245
253
309
341
365
373
381
405
429
437
445
461
469
477
485
493
501
509
629
693
725
749
757
765
821
853
877
885
893
917
941
949
957
973
981
989
997
1005
1013
1021
1205
1237
1261
1277
1397
1461
1493
1517
1525
1533
1653
1717
1749
1773
1781
1789
1845
1877
1901
1909
1917
1941
1965
1973
1981
1997
2005
2013
2021
2029
2037
2045
2229
2285
2485
2517

 

 

Written by meditationatae

September 23, 2017 at 7:01 am

Posted in History