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doubt some validity in this, i.e. proving the conjecture most likely involves getting
some grip on exactly when 1 )( = nT k . However, it doesn’t seem to follow that one will need to say exactly what the value of k is for any n , one only need to show that there is a k . Similarly, it seems that one cannot argue that from the )( nT k involving information about all of the intermediate numbers, e.g. the formula from Lagarais’s article, to it not being possible for there to be a shorter formula. For example, the formula for the sum ... − ++ + + n ar ar ar a . It doesn’t follow that there isn’t a simpler formula not involving the intermediate fact that there is a formula for of the first n terms of a geometric series can be written 1 2 Those are some initial thoughts on the proof. Feinstein would no doubt come back with counter arguments – as Chaitin’s work shows, while basing arguments on ideas such as ‘how much information is stored in a proof ’ is totally alien to any standard mathematical techniques, there is a new branch of mathematics which takes this seriously and is producing interesting results. The final word should perhaps go to Chaitin. He makes the point in his book ‘Metamaths: the quest for Omega’ [4] that while he believes that the implications for number theory of finding a fully random number (in his sense), Ω, is that there will always be true mathematical facts which cannot be proved, he is not certain that it is possible to show exactly which facts they are. He also tells the story of how, for many years, when asked for a likely example of a true but unprovable theorem, he would reply ‘Fermat’s Last Theorem’, adding how since 1995 and Andrew Wiles he has had to change his tune. numbers, in this case 1 r ra n − − )1 ( doing the trick.
References
[1] Lagarias, J.C. (1985) The 3x + 1 problem and its generalizations, Amer. Math. Monthly 92 (1985) 3-23, http://www.cecm.sfu.ca/organics/papers
[2] Feinstein, C.A. (2003), The Collatz 3n+1 Conjecture is Unprovable, http://arxiv.org/abs/math/0312309
[3] Turing, A.M. (1936) 'On computable numbers, with an application to the Entscheidungsproblem', http://www.turingarchive.org/browse.php/B/12
[4] Chaitin, G.J. (2006) Metamaths: The Quest for Omega, Vintage
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