If calculate directly with odd through $(\times 3+1)÷2^k$ operation, the odd sequence built (called Sequence (1)) has no obvious converge regularity, elements in the sequence vary sometimes big, sometimes small. But if do operation as introduced in above section, we would find convergence regularity of the odd sequence built (called Sequence (2)) is more obvious.

If add two corresponding elements in each step in these two odd sequences, should be exactly 2^k(k is different with different elements). Such as

$7+9=16,11+21=32,17+47=64$… in above example.

In general, first element in Sequence (1) can be written as(a_m-2…a₀=0, 1 or 2, a_m-1=-2, -1, 0, 1 or 2):

First element in Sequence (2) is: and, then

$x+a=3^m+3^{m-1}+3^{m-2}\times 2^2...+2^{2(m-1)}=2^{2m}$, is just the same form with Formula (2), and 2m should be the MSB+1 bit sequence number of x or a(along with the increase of a in Sequence (2), 2m should be the MSB+1 bit sequence number of a, because each corresponding part in Formula (2) is bigger than which in Formula (1). In fact we can select first a>x manually as long as sum of these two odds is 2^k).

Below prove next elements also satisfy above regularity.

Suppose a in Sequence (2) and x in Sequence (1) satisfy above regularity, and:

This states that the lowest bit of odd part of (3x+1) and (3a+2^m+1-1) is equal, add these two odd parts should be 2ⁱ(i<k).

Through above introduction we know, with odd do $(\times 3+1)÷2^k$ operation in the Collatz Conjecture odd sequence, equivalently, with the corresponding odd do $(\times 3+2^m-1)÷2^k$ using above iteration method. We can easily prove that odd 1…10a(a is in binary base) is equivalent to odd 10a in second method, count of succession 1 bits in the head part only represent the iteration steps roughly.

Build a simple weight model:

$w_i=\frac{{\mathrm{value}\mathrm{of}\mathrm{all}0\mathrm{bits}\mathrm{in}\mathrm{odd}\mathrm{part}\mathrm{in}\mathrm{t}}_{\mathrm{i}}}{2^{2\mathrm{k}}}$ Definition (1)

Which 2^2k is corresponding addition part in t_i in its step(we can also use the sum of t_i and its corresponding part in original sequence $2^{2k}$or $2^{2k+1}$ as denominator). Simply we can use w_i to represent the weight of value of all 0 bits in odd part in t_i. Specially, with any odd a, which highest bit is 2^m, define w_i for this odd:

$w_{[a]}=\frac{\mathrm{value}\mathrm{of}\mathrm{all}0\mathrm{bits}\mathrm{in}\mathrm{odd}\mathrm{a}}{2^{\mathrm{m}+1}}$ Definition (2)

At this time, use MSB+1 bit of each odd as 2^m in $(\times 3+2^m-1)÷2^k$ operation in each step. Although the denominator may be bigger than which in Definition (1), the regularity is same.

t_i sequence in above example is: 9,42,188,816,3456,14336

odd part sequence is: 9,21,47,51,27,7

w_i sequence is(according to Definition (1)):

(2+4)/4=1.5,(4+16)/16=1.25,64/64=1,(64+128)/256=0.75,512/1024=0.5,0/4096=0

Below prove w_i monotonically decreases except some special cases.

In fact, only one non-convergence case 0 bits in t_i do not shift right or bit-count reduce when t_i has not converged. This is:

101->1011.

This case w_i do not change, both are 1/4, according to Definition (2). But next step 1011->11 , t_i converges, hence this case is not worth worrying about.

Suppose with odd a do $(\times 3+1)÷2^k$ operation, and use x represent iteration steps. Reform w_i as following(according to Definition (1)), the numerator part is exactly equal to 0 bits in t_i:

Obviously w(x) is continuous derivable when a is in odd domain definition and x is in positive integer domain definition, and is bounded(>=0).

Now try to take the derivative of w(x).

Here the derivation definition of the numerator and denominator is: (y(x+1)-y(x))/(x+1-x).

Then the derivation of the numerator is:

The derivation of the denominator is: $2^{2k_x+2}-2^{2k_x}=3\times 2^{2k_x}$

Then

Which b is the odd after odd a doing x steps $(\times 3+1)÷2^k$ operation. that is:

Observe w^'(x), when b>3, w^'(x)<0, w(x) monotonically decreases. Only when b=1(this case $2^{p_{x+1}}$should be equal to 4), or when b=3, $2^{p_{x+1}}=2$, w^'(x)=0. Second case of b=3 is the exception case introduced above, the corresponding odd part of t_i is with form '101', is not worth worrying about. First case is convergence case.

Totally, this kind of iteration calculation has these cases after doing $(\times 3+2^m-1)÷2^k$. as following:

Case 1: odd tail part decreases one bit, head part does not increase one bit, this case tail part should insert one bit of 1 and with zero or more 0 changing to 1, totally 1 bits weight should increase in tail part.

Case 2: odd tail part decreases one bit, head part increases one bit, if corresponding odd in $(\times 3+1)÷2^k$sequence changes bigger, is just because tail part carries one bit of 1 to head part.

Case 3: odd tail part decreases two bits, head part does not increase one bit, tail part 0 bits should shift right.

Case 4: odd tail part decreases two bits, head part increases one bit.

Case 5: odd tail part decreases three or more bits, head part increases zero or one bit.

To avoid proving weight function W_[a] converging to 0(it is not easily to prove strictly the numerator part must be equal to 0 finally), build its complement weight function. Build:

$w_{c[a]}=\frac{\mathrm{a}}{2^{\mathrm{m}+1}}$, the highest bit of a is 2^m.

Through the proof and introduction above, we know W_c[a] monotonically increases except when corresponding odd b_i in $(\times 3+1)÷2^k$ sequence of a_i is 1 or 3, and these exception cases are not worth worrying about. And we also know the convergence state of W_c[a] is $\frac{2^{\mathrm{k}}-1}{2^{\mathrm{k}}}$.

Suppose odd a₀, a₁, a₂ are three elements in order in $(\times 3+2^m-1)÷2^k$ sequence, a₀ is equal to a, then

$w_{c[a_0]}=\frac{\mathrm{a}}{2^{\mathrm{m}+1}}$, $w_{c[a_1]}=\frac{3\mathrm{a}+2^{\mathrm{m}+1}-1}{2^{\mathrm{m}+3}}$, $w_{c[a_2]}=\frac{3^2\mathrm{a}+3\times 2^{\mathrm{m}+1}-3+2^{m+3}-2^p}{2^{\mathrm{m}+5}}$, where 2^p is 2^k in first step $(\times 3+2^m-1)÷2^k$ operation.

Observe this formula, when 2^p is equal to 2 or 4, $\frac{w_{c[a_2]}-w_{c[a_1]}}{w_{c[a_1]}-w_{c[a_0]}}$ is $\ge \frac{3}{4}$, suppose this ratio is $\frac{3}{4}$, then

When n->∞, $w_{c[a_n]}=\frac{\mathrm{a}}{2^{\mathrm{m}+1}}+\frac{2^{m+1}-a-1}{2^{m+3}}\times 4=\frac{2^{m+1}-1}{2^{\mathrm{m}+1}}$, this is a convergence state, in actual case, it needs a limit number n steps to reach to(or bigger than) $\frac{2^{m+1}-1}{2^{\mathrm{m}+1}}$, because the ratio is $\ge \frac{3}{4}$.

when 2^p is bigger than 4, $\frac{w_{c[a_2]}-w_{c[a_1]}}{w_{c[a_1]}-w_{c[a_0]}}$is $<\frac{3}{4}$, but still $>\frac{1}{2}$, W_c[a] also increases, W_c[a] can converge in $\frac{2^{\mathrm{k}}-1}{2^{\mathrm{k}}}$(k is any positive integer), not only $\frac{2^{\mathrm{m}+1}-1}{2^{\mathrm{m}+1}}$. This increases the convergence chance of W_c[a].

Observe the varying of fraction in lowest terms of W_c[a], the denominator part is equal, smaller, or 2 times of previous(because the numerator part at least can be divided by 2 in each step) in each step, when is equal, the numerator part should increase, it is possible to converge, when is 2 times of previous, the total value also increase, when is smaller, the total value should not only be bigger than the value of front W_c[a] with same denominator part(if exists), but also be bigger than all W_c[a] follow it. And in long sequence, usually appear smaller case, it has many chances to appear $\frac{2^{\mathrm{k}}-1}{2^{\mathrm{k}}}$, especially when the front element is already close to its convergence state. For example, suppose 177/256 is in sequence, if some following element with same denominator part 256 appear after many steps, its value should be bigger than all the elements between 177/256 and itself, it is much possible to be equal to 255/256.

Continuously observe W_c[a], even in the 2 times case, elements are closer to convergence state by themselves. Suppose the denominator part of fraction in lowest terms of $w_{c[a_1]}=\frac{3\mathrm{a}+2^{\mathrm{m}+1}-1}{2^{\mathrm{m}+3}}$ is 2^m+2,

2^m<a<2^m+1-1, if a is not equal to 11…101, which is very close to its convergence state 11…1, the above formula is <0. Thus prove the above conclusion.

Below give an example of start number 27 in $(\times 3+1)÷2^k$ odd sequence to verify, some decimals are written in the form which is easily to be judged equal to, bigger or smaller than 0.75.

Odds in $(\times 3+2^m-1)÷2^k$ sequence are:

37,87,97,209,441,917,1887,1927,1957,3959,3993,8037,16151,16209,32505,65141,130479,130627,65369,130821,261767,261861,523863,523969,1048097,2096433,4193225,8386989,16774787,8387697,16775849,33552381,67105787,16776639,16776783,16776891,4194243,2097129,4194269,8388555,1048571,262143

W_c[a] sequence:

37/64,87/128,97/128,209/256,441/512,917/1024,1887/2048,1927/2048,1957/2048,3959/4096,3993/4096,8037/8192,16151/16384,16209/16384,32505/32768,65141/65536,130479/(65536*2),130627/(65536*2),65369/65536,130821/(65536*2),261767/(65536*4),261861/(65536*4),523863/(65536*8),523969/(65536*8),1048097/(65536*16),2096433/(65536*32),4193225/(65536*64),8386989/(65536*128),16774787/(65536*256),8387697/(65536*128),16775849/(65536*256),33552381/(65536*512),67105787/(65536*1024),16776639/(65536*256),16776783/(65536*256),16776891/(65536*256),4194243/(65536*64),2097129/(65536*32),4194269/(65536*64),8388555/(65536*128),1048571/(65536*16),262143/262144

$w_{c[a_{i+1}]}-w_{c[a_i]}$ sequence:

13/128,10/128,15/256,23/512,35/1024,53/2048,40/2048,30/2048,45/4096,34/4096,51/8192,77/16384,58/16384,87/32768,131/65536,197/(65536*2),148/(65536*2),111/(65536*2),83/(65536*2),125/(65536*4),94/(65536*4),141/(65536*8),106/(65536*8),159/(65536*16),239/(65536*32),359/(65536*64),539/(65536*128),809/(65536*256),607/(65536*256),455/(65536*256),683/(65536*512),1025/(65536*1024),769/(65536*1024),144/(65536*256),108/(65536*256),81/(65536*256),15/(65536*64),11/(65536*64),17/(65536*128),13/(65536*128),1/(65536*16)

$\frac{w_{c[a_{i+2}}]-w_{c[a_{i+1}]}}{w_{c[a_{i+1}]}-w_{c[a_i]}}$ sequence:

10/13≈0.77,0.75,0.77,0.76,0.76,0.755,0.75,0.75,0.76,0.75,0.755,0.753,0.75,0.753,0.752,0.751,0.75,0.748,0.753,0.752,0.75,0.752,0.75,0.752,0.751,0.751,0.750,0.750,0.749,0.751,0.750,0.750,0.749,0.75,0.75,0.741,0.73,0.77,0.76,0.62

Through above we know$w_{c[a_1]}=\frac{3\mathrm{a}+2^{\mathrm{m}+1}-1}{2^{\mathrm{m}+3}}$, it can be written in following forms:

$w_{c[a_1]}=\frac{2\mathrm{a}+[\frac{b-1}{2}]}{2^{\mathrm{m}+2}}$, b-1<>0 mod 4, or

$w_{c[a_1]}=\frac{\mathrm{a}+[\frac{b-1}{4}]}{2^{\mathrm{m}+1}}$, b-1≡0 mod 4, in which b is the corresponding odd of a in $(\times 3+1)÷2^k$ sequence, b-1 reflects the 0-bits in the tail part of a.

Then Collatz Conjecture can be described as: With any odd a in range of 2^k to 2^k+1-1, set its initial goal set is 2^j+1-1(j<=k), its tail part is b, do operation: try to do (b-1) divided by 4, if can not, shift left one bit of a, plus the result of shifting right one bit of b(the 0-bits in the tail part of a), and add 2^k+2-1 to goals set of a, this operation makes the 0-bits in the tail part of a shift right or count reduce; if can, a plus the result of (b-1) divided by 4, this operation not only makes the 0-bits in the tail part of a shift right or count reduce, but also reduces the odds count about 1/4 to its goal 2^k+1-1, furthermore, if the last result is even, it can reduce a fraction of using 2^k+1 as denominator, this makes it can reach its previous goal 2^j+1-1(j<=k) possibly. Do these operations repeatedly, it has unlimited chances to reach to one of its goal set.

Through above we know, if $(\times 3+1)÷2^k$ sequence have only /2 and(or) /4 cases, the sequence can never converge, /2 case makes goal of a in $(\times 3+2^m-1)÷2^k$ sequence larger, /4 case needs ∞ steps. But it is not possible in long sequence, this is determined by the regularity of tail binary bits of odd doing $(\times 3+1)÷2^k$operation. Odds of form *10…01(many 0), result can do /4, Odds of form *11…11(many 1), result can do /2, these two cases must become other forms after several steps. Odds with other forms, themselves and their following steps must appear alternately /2, /4, /2^k(k>2) cases.

6. (*3+2^. m-1)/2^k Odd Tree and Its Convergence Regularity

6.1. (*3+2^m-1)/2^k Odd Tree and Its Characters

We call 2^k are the properties of odds after doing $(\times 3+2^m-1)÷2^k$operation. See following tree:

…

L6: 129(321.1) 131(81.3) 133(327.1) 135(165.2) 137(333.1) 139(21.5) 141(339.1) 143(171.2) 145(345.1) 147(87.3) 149(351.1) 151(177.2) 153(357.1) 155(45.4) 157(363.1) 159(183.2) 161(369.1) 163(93.3) 165(375.1) 167(189.2) 169(381.1) 171(3.8) 173(387.1) 175(195.2) 177(393.1) 179(99.3) 181(399.1) 183(201.2) 185(405.1) 187(51.4) 189(411.1) 191(207.2) 193(417.1) 195(105.3) 197(423.1) 199(213.2) 201(429.1) 203(27.5) 205(435.1) 207(219.2) 209(441.1) 211(111.3) 213(447.1) 215(225.2) 217(453.1) 219(57.4) 221(459.1) 223(231.2) 225(465.1) 227(117.3) 229(471.1) 231(237.2) 233(477.1) 235(15.6) 237(483.1) 239(243.2) 241(489.1) 243(123.3) 245(495.1) 247(249.2) 249(501.1) 251(63.4) 253(507.1) 255

L5: 65(161.1) 67(41.3) 69(167.1) 71(85.2) 73(173.1) 75(11.5) 77(179.1) 79(91.2) 81(185.1) 83(47.3) 85(191.1) 87(97.2) 89(197.1) 91(25.4) 93(203.1) 95(103.2) 97(209.1) 99(53.3) 101(215.1) 103(109.2) 105(221.1) 107(7.6) 109(227.1) 111(115.2) 113(233.1) 115(59.3) 117(239.1) 119(121.2) 121(245.1) 123(31.4) 125(251.1) 127

L4: 33(81.1) 35(21.3) 37(87.1) 39(45.2) 41(93.1) 43(3.6) 45(99.1) 47(51.2) 49(105.1) 51(27.3) 53(111.1) 55(57.2) 57(117.1) 59(15.4) 61(123.1) 63

L3: 17(41.1) 19(11.3) 21(47.1) 23(25.2) 25(53.1) 27(7.4) 29(59.1) 31

L2: 9(21.1) 11(3.4) 13(27.1) 15

L1: 5(11.1) 7

L0: 3

In above tree, a.b in () means result is a*2^b after front odd doing $(\times 3+2^m-1)÷2^k$ operation, m_th layer has 2^m elements, the last element is the convergence state. Characters of 2^k are very regular, for example, upward from a specific layer, positions of 2 are 1+2i(i>=0), upward from another specific layer, positions of 2² are 4+4i, positions of 2³ are 2+8i, positions of 2⁴ are 14+16i…, this can be easily proved strictly. For example, odds of position 2+8i in m layer are 2^m+1-1+(2+8i)*2, (0=<i<=[(2^m-1-1)/4]).

Can be divided by 2³, result is odd if m+1>3. And because the highest bit of the result odd is 2^m, it must be in m-1 layer, downward one layer from m layer.

Through above, we can easily prove that if the property of an odd is 2¹, it moves upward one layer(and also moves forward some location), if the property of an odd is 2², it moves forward in the same layer, if the property of an odd is 2^k(k>2), it moves downward k-2 layers(and also moves forward some location).

In this tree, because element count of each layer is 2 times of which of the downward layer, we can transform all positions to one specific layer, m-2 layer transform to m-1 layer do $\times 2$, m layer transform to m-1 layer do $÷2$, etc. Then all transform positions(to m-1 layer) can not exceed 2^m-1!

Next try to prove odds in any layer can converge. We know loop odd sequence and divergence odd sequence both are long non-convergence sequence(expand loop sequence). Below suppose the research sequence is long huge(odds in $(\times 3+1)÷2^k$ sequence are huge) sequence.

Suppose a is an odd in m-1 layer, its highest bit is 2^m.

Position of a in m-1 layer is: $\frac{a-2^m+1}{2}$,

$3\times a+2^{m+1}-1=b\times 2^{p_1}$, b is in layer m-p₁+1

Position of b in m-p₁+1 layer is: $\frac{b-2^{m-p_1+2}+1}{2}$,

Position of b in m-1 layer is: $\frac{b-2^{m-p_1+2}+1}{2^{3-p_1}}$

$3^2\times a+3\times 2^{m+1}-3+2^{m+3}-2^{p_1}=c\times 2^{p_1+p_2}$, c is in layer m+3-p₁-p₂

Position of c in m+3-p₁-p₂ layer is: $\frac{c-2^{m+4-p_1-p_2}+1}{2}$

Position of c in m-1 layer is: $\frac{c-2^{m+4-p_1-p_2}+1}{2^{5-p_1-p_2}}$

$\frac{c-2^{m+4-p_1-p_2}+1}{2^{5-p_1-p_2}}-\frac{b-2^{m-p_1+2}+1}{2^{3-p_1}}=\frac{c+1-2^{2-p_2}\times b-2^{2-p_2}}{2^{5-p_1-p_2}}$, ratio p is:

Because in long huge sequence, 2^m+1-a is very big, the ratio p is very close to 3/4. Only these cases ratio p<3/4: p₂=1, p₁>=2; p₂=2, p₁>=3. This is to say: cases of (forward, upward), (downward, upward), (downward, forward) ratio <3/4; cases of (upward, upward), (downward, downward), (upward, downward), (upward, forward), (forward, downward) ratio >3/4; case of (forward, forward) ratio =3/4. Obviously, the ratio regularity is also suitable for all following steps.

We can deduce the common transform position formula:

which b_i is the corresponding odd in $(\times 3+1)÷2^k$ sequence.

Next try to rebuild a new sequence based on the original sequence which average value of transform position increment ratio p is >3/4.

Suppose one long huge sequence has n upward steps, k forward steps, l downward steps, finally upward h layers, n>h and n>l.

The final transform position is:

Now rebuild, do not change the first step, move number property 2^k of all downward steps next to the first step(suppose we can move in order to compare, we can do it from math calculation angle), then forward steps and upward steps. Watch common transform position formula, since $2^{p_1+p_2+...p_{n+k+l}}$ and $2^{2(n+k+l)+1}$are not changed, the new sequence must have smaller final “transform position”(virtual transform position) than original because more previous the position of number property 2^k is, more great influence it produces, bigger the value of subtractive part is.

For example

Hence can use new sequence to estimate the convergence of the original sequence

Then merge the upward steps in the end, forward steps and the last downward step before forward steps to one step. The new sequence(has l or l+1 steps) must have a “transform position” increment ratio>3/4 according to the ratio formula which indicates that (upward, downward), (forward, downward), (downward, downward) have a transform position increment ratio>3/4

If long huge sequence is non-convergence, it must appear downward steps continuously after some upward and/or forward steps each time, the count of downward steps must be infinite. So we can use proportional sequence of ratio 3/4 to estimate the rebuilt sequence.

After a do z(=l or l+1) times $(\times 3+2^m-1)÷2^k$ operation, “transform position” is:

When z->∞,