bound for the smallest term of a -cycle: for example, if there is a 77-cycle, then at least one element of the cycle is less than . Along with the verification of the conjecture up to , this implies the nonexistence of a non-trivial -cycle up to . As exhaustive computer searches continue, larger values may be ruled out. To state the argument more intuitively: we need not look for cycles that have at most 77 circuits, where each circuit consists of consecutive ups followed by consecutive downs. Other formulations of the conjecture In reverse There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph. The Collatz graph is a graph defined by the inverse relation So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer , if and only if . Equivalently, if and only if . Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function defined in the Statement of the problem section of this article). When the relation of the function is replaced by the common substitute "shortcut" relation , the Collatz graph is defined by the inverse relation, For any integer , if and only if . Equivalently, if and only if . Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above). Alternatively, replace the with where and is the highest power of 2 that divides (with no remainder). The resulting function maps from odd numbers to odd numbers. Now suppose that for some odd number , applying this operation times yields the number 1 (that is, ). Then in binary, the number can be written as the concatenation of strings where each is a finite and contiguous extract from the representation of . The representation of therefore holds the repetends of , where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs. Conjecturally, every binary string that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to ). As an abstract machine