Collatz-Intro - The primitive loop
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Workshop |
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Introdruction
to the "primitive loop"
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The term
"primitive loop" is a private notion, 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- only 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 |
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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. |
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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]. |
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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 |
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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 equation and we from here 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 |
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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) |
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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) |
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last update: 15.8.2004 |