Tag Archives: number theory

Every odd positive integer is connected to 1

The elements at the ‘horizon’ of the diagram are the irrational elements >1 in the extension field Q[sqrt5] encoded in binary in the Stern-Brocot tree. In that tree between every two integer values to the right of 1 there is a subtree isomorphic to the whole left side (the subtree between 0 and 1). As encodedContinue reading “Every odd positive integer is connected to 1”

A perfect binary Collatz tree

To compute the OPI values in the tree…Let q = the quotient of the lower OPI and r = its residue mod 3: These rules will generate the identical OPI not ≡ 0 mod 3 found in the graph generated by the original Collatz function and will map onto the odd positive integer results ofContinue reading “A perfect binary Collatz tree”