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

Posted: Jun 3, 2009 9:42 PM
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Musatov left a note: P=NP "Now I have a machine gun."

Mensanator wrote:
> 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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