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Unformatted text preview: arXiv:math/0312309v16 [math.GM] 4 May 2006 The Collatz 3n+1 Conjecture is Unprovable Craig Alan Feinstein May 4, 2006 Baltimore, Maryland U.S.A. email: [email protected], BS”D Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. In this note, we consider the following function: Definition 1: Let T : N → N be a function such that T ( n ) = 3 n +1 2 if n is odd and T ( n ) = n 2 if n is even. The Collatz 3 n + 1 Conjecture states that for each n ∈ N , there exists a k ∈ N such that T ( k ) ( n ) = 1, where T ( k ) ( n ) is the function T iteratively applied k times to n . As of September 4, 2003, the conjecture has been verified for all positive integers up to 224 × 2 50 ≈ 2 . 52 × 10 17 (Roosendaal, 2003+). Furthermore, one can give a heuristic probabilistic argument (Crandall, 1978) that since every iterate of the function T decreases on average by a multiplicative factor of about ( 3 2 ) 1 / 2 ( 1 2 ) 1 / 2 = ( 3 4 ) 1 / 2 , all iterates will eventually converge into the infinite cycle { 1 , 2 , 1 , 2 , ... } , assuming that each T ( i ) sufficiently mixes up n as if each T ( i ) ( n ) mod 2 were drawn at random from the set { , 1 } . However, the Collatz 3 n + 1 Conjecture has never been formally proven. In this paper, we show using Chaitin’s notion of randomness (Chaitin, 1990) that the Collatz 3 n + 1 Conjecture can, in fact, never be formally proven, even though there is a lot of evidence for its truth. The underlying assumption in our argument is that a proof is composed of bits (zeroes and ones) just like any computer textfile. First, let us present a definition of “random”. Definition 2: We shall say that vector x ∈ { , 1 } k is random if x cannot be specified in less than k bits in a computer textfile. For instance, the vector x = [010101 ... 010101] ∈ { , 1 } 10 6 is not random, since we can specify x in less than one million bits in a computer textfile. (We just did.) However, the vector of outcomes of one million cointosses has a good chance of fitting our definition of “random”, since much of the time the most compact way of specifying such a vector in a computer textfile is to list the results of each cointoss, in which one million bits are necessary. Theorem 1: For any vector x ∈ { , 1 } k , there exists an n ∈ N such that x = ( n, T ( n ) , ..., T ( k 1) ( n )) mod 2....
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This note was uploaded on 12/28/2011 for the course ECE 1 taught by Professor Chen during the Fall '08 term at UCSB.
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