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*                                         
*  Length 3 Theorems for bases             
*         {4, 7, 9, 10, 14, 16}           
*                                         
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For a given base b and number of digits l, let N(l,b) denote the number of
limiting orbits occuring over all possible input numbers num with l digits
in base b to the iterative procedure listPhenom(num,b), and let L(l,b) denote
the list of lengths of each orbit, sorted in increasing order.


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Odd-Base Conjecture(s)

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For an odd b >= 3, let n be the unique integer s.t. b = 2*n+1.
Then we have the following theorem:

N(3,b) = 1. Specifically, L(3,b) = [2].
In particular, the single limiting orbit is given by [[n-1, 2*n, n+1], [n, 2*n, n]]. 



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  base b = 7  
*****************

For b = 7, n = 3.
The list of all limiting orbits is: 


                     [[2, 6, 4], [3, 6, 3]]

Thus N(3,b) = 1 and L(3,b) = [2].
Further, the limiting orbit is [[2, 6, 4], [3, 6, 3]].
Thus the conjecture is true.

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  base b = 9  
*****************

For b = 9, n = 4.
The list of all limiting orbits is: 


                     [[3, 8, 5], [4, 8, 4]]

Thus N(3,b) = 1 and L(3,b) = [2].
Further, the limiting orbit is [[3, 8, 5], [4, 8, 4]].
Thus the conjecture is true.

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Even-Base Conjecture(s)

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For an even b >= 2, let n be the unique integer s.t. b = 2*n
Then we have the following theorem:

Theorem: N(3,b) = 1. Specifically, L(3,b) = [1].
In particular, the single limiting orbit is given by [[n-1, 2*n-1, n]].


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  base b = 4  
*****************

For b = 4, n = 2.
The list of all limiting orbits is: 


                          [[1, 3, 2]]

Thus N(3,b) = 1 and L(3,b) = [1].
Further, the limiting orbit is [[1, 3, 2]].
Thus the conjecture is true.

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  base b = 10  
*****************

For b = 10, n = 5.
The list of all limiting orbits is: 


                          [[4, 9, 5]]

Thus N(3,b) = 1 and L(3,b) = [1].
Further, the limiting orbit is [[4, 9, 5]].
Thus the conjecture is true.

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  base b = 14  
*****************

For b = 14, n = 7.
The list of all limiting orbits is: 


                          [[6, 13, 7]]

Thus N(3,b) = 1 and L(3,b) = [1].
Further, the limiting orbit is [[6, 13, 7]].
Thus the conjecture is true.

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  base b = 16  
*****************

For b = 16, n = 8.
The list of all limiting orbits is: 


                          [[7, 15, 8]]

Thus N(3,b) = 1 and L(3,b) = [1].
Further, the limiting orbit is [[7, 15, 8]].
Thus the conjecture is true.


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*                                        
*  Length 4 conjecture(s) for bases       
*         {4, 7, 9, 10, 14, 16}           
*                                         
*******************************************

For a given base b and number of digits l, let N(l,b) denote the number of
limiting orbits occuring over all possible input numbers num with l digits
in base b to the iterative procedure listPhenom(num,b), and let L(l,b) denote
the list of lengths of each orbit, sorted in increasing order.


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Odd-Base Conjecture(s)

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For an odd b >= 5, let n be the unique integer s.t. b = 2*n+1.
Finally, let U be the union of all the 4-digit numbers which occur in some limiting orbit.

Then we have the following conjectures:

Conjecture A (Parity Conjecture): If n mod 4 = 0 or 1 then N(4,b) is even, and if n mod 4 = 2 or 3, then N(4,b) is odd.

Conjecture B (Orbit Lengths Division Conjecture): Each integer in L(4,b) divides m.

Conjecture C (Elements Conjecture): U = {seq(seq([2*k+1, 2*i, 2*(n-i)-1, 2*(n-k)], i=0..k-1), k=1..n-1)}.


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  base b = 7  
*****************

For b = 7, n = 3, and the list of all limiting orbits is: 


           [[3, 0, 5, 4], [5, 0, 5, 2], [5, 2, 3, 2]]

Thus N(4,b) = 1, n mod 4 = 3, L(4,b) = {3}, and m = 3.
Since N(4,b) has odd parity, Conjecture A is true.
and by checking each entry of L(4,b) we see that Conjecture B is true. 
Finally, in this case U = {seq(seq([2*k+1, 2*i, 5-2*i, 6-2*k], i=0..k-1), k=1..2)}, which more explicitly is:

           {[3, 0, 5, 4], [5, 0, 5, 2], [5, 2, 3, 2]}

and so comparing U with the list of limiting orbits above, we see that in this case Conjecture C is true.

*****************
  base b = 9  
*****************

For b = 9, n = 4, and the list of all limiting orbits is: 


           [[3, 0, 7, 6], [7, 2, 5, 2], [5, 2, 5, 4]]
           [[5, 0, 7, 4], [7, 0, 7, 2], [7, 4, 3, 2]]

Thus N(4,b) = 2, n mod 4 = 0, L(4,b) = {3}, and m = 3.
Since N(4,b) has even parity, Conjecture A is true.
and by checking each entry of L(4,b) we see that Conjecture B is true. 
Finally, in this case U = {seq(seq([2*k+1, 2*i, 7-2*i, 8-2*k], i=0..k-1), k=1..3)}, which more explicitly is:

   {[3, 0, 7, 6], [5, 0, 7, 4], [5, 2, 5, 4], [7, 0, 7, 2], 

     [7, 2, 5, 2], [7, 4, 3, 2]}

and so comparing U with the list of limiting orbits above, we see that in this case Conjecture C is true.

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2-Power Conjectures

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For a b >= 4 which is a power of 2, let n be the unique integer s.t. b = 2^n.Then we have the following conjectures:

n-even conjecture: N(4,b) = n. Specifically, writing n = 2k we have that L(4,b) = [k, k+1, seq(2k, i=1..k-1), seq(2(k+1), i=1..k-1)].

n-odd conjecture: N(4,b) = n-1. Specifically, writing n = 2k+1 we have that L(4,b) = [seq(2k+1, i=1..k), seq(2(k+1)+1, i=1..k)]


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  base b = 4  
*****************

For b = 4, so n = 2, and in particular n is even, and k = 1.
The list of all limiting orbits is: 


                         [[3, 0, 2, 1]]
                  [[1, 3, 3, 2], [2, 0, 2, 2]]

Thus N(4,b) = 2 and L(4,b) = [1, 2].
Thus the n-even Conjecture is true.

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  base b = 16  
*****************

For b = 16, so n = 4, and in particular n is even, and k = 2.
The list of all limiting orbits is: 


                [[5, 2, 12, 11], [10, 5, 9, 6]]
        [[3, 15, 15, 12], [12, 2, 12, 4], [10, 7, 7, 6]]
 [[3, 0, 14, 13], [14, 9, 5, 2], [12, 3, 11, 4], [9, 6, 8, 7]]
[[1, 15, 15, 14], [14, 0, 14, 2], [14, 11, 3, 2], [12, 7, 7, 4], 

  [7, 15, 15, 8], [8, 6, 8, 8]]

Thus N(4,b) = 4 and L(4,b) = [2, 3, 4, 6].
Thus the n-even Conjecture is true.

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3-Power Conjectures

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For a base b >= 9 which is a power of 3, let n be the unique integer s.t. b = 3^n. 
Then we have the following conjecture:

Conjecture:  N(4,b) = sum(3^i-1, i=1..n-1). In particular L(4,b) = [seq(seq(3^k, i=1..3^k-1), k=1..n-1)].


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  base b = 9  
*****************

For b = 9, n = 2.
The list of all limiting orbits is: 


           [[3, 0, 7, 6], [7, 2, 5, 2], [5, 2, 5, 4]]
           [[5, 0, 7, 4], [7, 0, 7, 2], [7, 4, 3, 2]]

Thus N(4,b) = 2 and L(4,b) = [3, 3].
Thus the conjecture is true.

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5x2-Power Theorems

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*****************
  base b = 10  
*****************

For b = 10, n = 1, and in particular n mod 4 is 1.
The list of all limiting orbits is: 


                         [[6, 1, 7, 4]]

Thus N(4,b) = 1 and L(4,b) = [1].
Further, the limiting orbit is [[6, 1, 7, 4]].
Thus the n-odd Conjecture is true.

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7x2-Power Conjectures

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For a base b >= 14 which is of the form b = 7*2^n, we have the following conjectures:

n mod 3 = 0 conjecture: N(4,b) = 2. Specifically, writing n = 3k we have that L(4,b) = [3, 3k+1].
In particular, the first orbit is generated by [3*2^n, 2^n - 1, 3*2^(n+1) - 1, 2^(n+2)]
and the second orbit is generated by [2^n-1, 7*2^n-1, 7*2^n - 1, 3*2^(n+1)]. 

n mod 3 nonzero conjecture: N(4,b) = 1. Specifically, L(4,b) = [3].
In particular, the singular orbit is generated by [3*2^n, 2^n - 1, 3*2^(n+1) - 1, 2^(n+2)].


*****************
  base b = 14  
*****************

For b = 14, n = 1, and in particular n mod 3 = 1.The list of all limiting orbits is: 


         [[6, 1, 11, 8], [10, 1, 11, 4], [10, 5, 7, 4]]

Thus N(4,b) = 1 and L(4,b) = [3].
Further, the generator for the limiting orbit is [6, 1, 11, 8].
Thus the n mod 3 nonzero conjecture is true.


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