N.B. This dead slide dates from the days when the three tree types were
regarded as separate tree rather than three manifestations of a single
predecessor tree. These points are settled now.
Why Doesn't That Settle the Issue?
-- If all odd integers are in the abstract predecessor tree, doesn't that prove the conjecture?
return to slide index
-- It would, if it had been the general or the binary predecessor tree.
-- We lost it when we mapped the binary tree into the abstract tree.
-- That mapping works only in the one direction, from binary to abstract.
-- Placing an arbitrary integer in the abstract tree does not place it in the binary tree
because we don't have a mapping from the abstract to the binary tree.
-- In terms of the predecessor graph, if it were disconnected such
that an additional graph existed along side the binary tree, the
elements of that additional graph might be a tree composed of l.d.a.s
and rooted in something other than 1. Such a disconnected graph would constitute a
large family of counter-examples.
-- I.e., of all the values n might take in some particular dn+c
formulas, some number of them might specify l.d.a.s in counter-example subgraphs.