2 9 Se p 20 04 The Collatz 3 n + 1 Conjecture is Unprovable

The Collatz 3n+ 1 Conjecture states that for each n ∈ N, there exists an m ∈ N such that T (n) = 1, where T (n) is the function T iteratively applied m times to n. As of September 4, 2003, the conjecture has been verified for all positive integers up to 224× 2 ≈ 2.52× 10 (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 (32 ) (12 ) 1/2 = ( 4 ) , 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 (n) mod 2 were drawn at random from the set {0, 1}. However, the Collatz 3n+1 Conjecture has never been formally proven. In this paper, we show using Chaitin’s notion of randomness (Chaitin, 1990) that the Collatz 3n+ 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 text-file. First, let us present a definition of “random”.