Editing Collatz conjecture (section)

==Syracuse function==
If {{mvar|k}} is an odd integer, then {{math|3''k'' + 1}} is even, so {{math|3''k'' + 1 {{=}} 2<sup>''a''</sup>''k''{{prime}}}} with {{math|''k''{{prime}}}} odd and {{math|''a'' ≥ 1}}. The '''Syracuse function''' is the function {{mvar|f}} from the set {{mvar|I}} of positive odd integers into itself, for which {{math|''f''(''k'') {{=}} ''k''{{prime}}}} {{OEIS|id=A075677}}.

Some properties of the Syracuse function are:

* For all {{math|''k'' ∈ ''I''}}, {{math|''f''(4''k'' + 1) {{=}} ''f''(''k'')}}. (Because {{math|3(4''k'' + 1) + 1 {{=}} 12''k'' + 4 {{=}} 4(3''k'' + 1)}}.)
* In more generality: For all {{math|''p'' ≥ 1}} and odd {{mvar|h}}, {{math|''f''{{isup|''p'' − 1}}(2<sup>''p''</sup>''h'' − 1) {{=}} 2 × 3<sup>''p'' − 1</sup>''h'' − 1}}. (Here {{math|''f''{{isup|''p'' − 1}}}} is [[Functional power|function iteration notation]].)
* For all odd {{mvar|h}}, {{math|''f''(2''h'' − 1) ≤ {{sfrac|3''h'' − 1|2}}}}

The Collatz conjecture is equivalent to the statement that, for all {{mvar|k}} in {{mvar|I}}, there exists an integer {{math|''n'' ≥ 1}} such that {{math|''f''{{isup|''n''}}(''k'') {{=}} 1}}.