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Topic: Max. power of 2 and number of odd integers in a loop in the Collatz
problem

Replies: 7   Last Post: Jun 6, 2009 4:45 PM

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mensanator

Posts: 5,039
Registered: 12/6/04
Re: Max. power of 2 and number of odd integers in a loop in the
Collatz problem

Posted: Jun 3, 2009 1:41 PM
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On Jun 3, 4:43 am, roupam <roupam.gh...@gmail.com> wrote:
> I have found, that if there exists a loop, then if
> n = number of odd integers in the loop
> k = maximum power of 2 that divides a number in the loop
> then, the following conclusions hold...
> for positive integers
>    k <= n+1
> for negative integers
>    k < n log3//log2 - n + 1
> Consider the negative loop starting with
> -17 -50 -25 -74 -37 -110 -55 -164 -82 -41 -122 -61 -182 -91 -272 -136
> -68 -34 and then back to -17
> Here...
> n = 7
> k = 4
> 4 < 7 * log3/log2 -7 + 1 = 5.09...


What does this follow from, other than your observation?

It is certainly not true in 3n+C, so why does it have
to be true in 3n+1?

Suppose C is a power of 2 minus 3. Then there will ALWAYS
be a loop of exactly one odd number that has k even numbers
that loops at 1:

import collatz_functions as cf
C = 2**94 - 3
print 'C = ',C
sv = cf.build_sv(1,1,1000,C)
print sv
cf.zigzag(1,sv,C,7)


C = 19807040628566084398385987581
[94] # 1 odd number, 94 even numbers

1
19807040628566084398385987584
9903520314283042199192993792
4951760157141521099596496896
2475880078570760549798248448
1237940039285380274899124224
618970019642690137449562112
309485009821345068724781056
154742504910672534362390528
77371252455336267181195264
38685626227668133590597632
19342813113834066795298816
9671406556917033397649408
4835703278458516698824704
2417851639229258349412352
1208925819614629174706176
604462909807314587353088
302231454903657293676544
151115727451828646838272
75557863725914323419136
37778931862957161709568
18889465931478580854784
9444732965739290427392
4722366482869645213696
2361183241434822606848
1180591620717411303424
590295810358705651712
295147905179352825856
147573952589676412928
73786976294838206464
36893488147419103232
18446744073709551616
9223372036854775808
4611686018427387904
2305843009213693952
1152921504606846976
576460752303423488
288230376151711744
144115188075855872
72057594037927936
36028797018963968
18014398509481984
9007199254740992
4503599627370496
2251799813685248
1125899906842624
562949953421312
281474976710656
140737488355328
70368744177664
35184372088832
17592186044416
8796093022208
4398046511104
2199023255552
1099511627776
549755813888
274877906944
137438953472
68719476736
34359738368
17179869184
8589934592
4294967296
2147483648
1073741824
536870912
268435456
134217728
67108864
33554432
16777216
8388608
4194304
2097152
1048576
524288
262144
131072
65536
32768
16384
8192
4096
2048
1024
512
256
128
64
32
16
8
4
2
1



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