Collatz Conjecture - Statement of The Problem

Statement of The Problem

Consider the following operation on an arbitrary positive integer:

  • If the number is even, divide it by two.
  • If the number is odd, triple it and add one.

In modular arithmetic notation, define the function f as follows:

Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.

In notation:

(that is: is the value of applied to recursively times; )

or

{a_{i}} = \frac{1}{2}{a_{i-1}} - \frac{1}{4}(5a_{i-1}+2)((-1)^{a_{i-1}}-1)

(which yields for even and for odd ).

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially.

That smallest i such that ai = 1 is called the total stopping time of n. The conjecture asserts that every n has a well-defined total stopping time. If, for some n, such an i doesn't exist, we say that n has infinite total stopping time and the conjecture is false.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1. Such a sequence might enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.

Read more about this topic:  Collatz Conjecture

Famous quotes containing the words statement of the, statement of, statement and/or problem:

    “Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
    —Mario Vargas Llosa (b. 1936)

    “One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
    —Rebecca West (1892–1983)

    “Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
    —William Burroughs (b. 1914)

    “We have heard all of our lives how, after the Civil War was over, the South went back to straighten itself out and make a living again. It was for many years a voiceless part of the government. The balance of power moved away from it—to the north and the east. The problems of the north and the east became the big problem of the country and nobody paid much attention to the economic unbalance the South had left as its only choice.”
    —Lyndon Baines Johnson (1908–1973)