Collatz-Intro - The primitive loop
|
Workshop |
|
Introdruction to the "primitive loop"
|
The term "primitive loop" is a private invention, resulting from the problem, that the analysis of the general loop was too complicated to derive decisive results. to keep the formulas simpler for a start, I decided to focus the simple form: some ascending steps- one descending step. I separate things in four parts: 2.2) modular reqirements for the least element in a primitve loop 2.3) the critical diophantine equation for the primitve loop 4.1) the combined critical diophantine equation and approximation inequality |
|
In my notation: primitive Transformation a' = T(a;1,1,1,1,...,1,A) with length N, N-1 ascending steps (all exponents are 1), 1 descending step (exponent>1). In newer writing I make use of the further abbreviation a' = PT(a;N:A) indicating: N-1 steps with exponent 1, 1 step with exponent A. From that : "primitive loop: a' = T(a;1,1,1,1,1,1...,1,A) = a or a' = PT(a;N:A) = a The PT-N-loop The next step of complexity is then to concatenate several "primitive transformations" into one: a' = PT( a;N1:A,N2:B,N3:C) or a = PT( a;N1:A,N2:B,N3:C) forming a concatenated-primitive-loop (PT-N-Loop) One can see, that this -with lengthes N=1 - can as well describe the general loop: then there is no ascending step between two descending steps, and thus an arbitrary concatenation of primitive transformations allowing N=1 can be seen as a proper description of a general loop. |
|
Note: an analoguous concept was followed by several authors, for instance [Steiner],[de Weger],[Schorer] with the notation "1-cycle", where one step is defined as either b = (3a+1)/2 if a odd or b = a/2 if a even. Thus the parameters of the "1-cycle" K,L indicating the numbers of ascending resp descending steps relate to my notation: 1-cycle(K,L) equivalent a = PT(a;K:L+1) = P(a;N:A) The equivalent generalization to concatenation of more of one "1-cycles" (which are no more cycles, btw) is the notation: "m-cycle" in [DeWeger]. |
|
To attack the problem for any length of a primitive loop, it is wise, to look deeper into the structure of an "a" which could be the first member of such a longer loop. We know already from the canonical eqution for transformations, that 3^N a' = T(a;A,B,C) = ----- a + T(0;A,B,C) ( with N exponents and S=sum-of-exponents) 2^S |
|
Now we let all exponents be 1, and denote the length of this purely ascending steps as L = N-1 For such a transformation of the length L also S equals L. For convenience of notation let z be the last member of such a repeated transformation then 3^L 3^L - 2^L z = T(a;1,1,1...1) = ---- a + ---------- 2^L 2^L 3^L 3^L - 2^L z = ---- a + ---------- 2^L 2^L z*2^L = 3^L ( a + 1) - 2^L // multiplying by 2^L
2^L (z+1) = 3^L (a + 1) // collecting powers of equal base so z + 1 a + 1 ------- = ------- thus all a of the structure a = k*2^L - 1 // with the free parameter k and z = k*3^L - 1 // with the same free parameter k satisfy this equation, because (k*3^L-1) + 1 (k*2^L-1) + 1 -------------- = -------------- k*3^L k*2^L --------------
= -------------- k = k all k satisfy the euqation and we introduce k as a free parameter. From the general approach, to use odd numbers>1 in a loop (but this restriction can as well been dismissed), we can apply some restrictions to this parameter k, so that it produces valid values for a and z. First we assume always, that all members a,b,c...z of a loop are odd and greater 1, so from z = k*3^L -1 it follows also, that k must be even to keep z odd, thus have the form 2i, thus z = 2*3^L*i -1 // replacing k by 2*i and a = 2*2^L*i - 1 // replacing k by 2*i Now add one decreasing transformation at the end, to make a loop possible a'= T(z;A) so z*3+1 (2*3^L*i - 1)*3 + 1 (2*3^L*i -
1)*3 + 1 then 2*3^N * i -
2 3^N*i - 1 finally the structure of a', depending on N and a free parameter i 3^N * i - 1 That means, for a primitive transformation a' = P(a;N:A) we know the required structure of a and a': A primitive transformation a' = T(a;1,1,1,1...1,A) with N steps and L=N-1 ones, in shorter notation a' = PT(a;N:A) requires as structures for input a and for output a' a = 2^N*i - 1 (from a = 2*2^L*i - 1 ) and 3^N *i - 1 2^(A-1) with the same free parameter i>0 Also we recognize that a', being odd, cannot be divisible by 3 (because the nominator of rhs is not divisible by 3) thus the output of such a transformation can be written as +1 |
|
To make the primitive transformation a primitive loop, the leading member "a" must match both equations, that for a and that for a', and we have a much more restrictive condition a primitive loop of the form a = PT(a;N:A) requires for the structure of the first member "a" 3^N *i - 1 2^(A-1) |
|
From this follows the critical diophantine equation for the primitive loop, which must be solvable in integers N,A, and i (N>A>1, i>0 ) 3^N*i -
1 (where the last condition on i excludes the unwanted case of a=-1). This result suggests two basic modular restrictions: 1) i must be odd to keep the nominator even and thus divisible by 2^(A-1) 2) a cannot be divisible by 3, but must have the form a = 6i+1 or 6i+5 (see example of one transformation) |
|
last update: 15.8.2004 |