## An Arbitrary Odd Integer Can Be Precisely Located in the Abstract Predecessor Tree

### -- Use *o* as the odd integer to be placed.

-- Run Collatz (upward in the predecessor tree) until an extension is reached.

-- Develop (downward in the predecessor tree) the l.d.a. under that
extension (clear to the leaf node) in 3 ways.

- (a) the 2^{a}*3^{b}**n*+*c* formulas
- (b) the values of *c*

- (c) the string representing the l.d.a., call it *p*

- *o* will be reached at some step in the l.d.a., call it *j*

- *o* might be the *e* element, the *t* element, or an internal element in the l.d.a.

-- equate a formula from (a) with a value of *c* from (b); solve for *n*

-- *o* is now characterized as the *j*th element of the *n*th instantiation of the l.d.a. *p*,
which locates it exactly in the abstract predecessor tree.

Since every odd integer can be located in the abstract
predecessor tree, every odd integer must appear in the abstract predecessor
tree.

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