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Mathematics of Computation

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Density bounds for the $ 3x+1$ problem. I. Tree-search method

Authors: David Applegate and Jeffrey C. Lagarias
Journal: Math. Comp. 64 (1995), 411-426
MSC: Primary 11B83; Secondary 11Y99
MathSciNet review: 1270612
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Abstract: The $ 3x + 1$ function $ T(x)$ takes the values $ (3x + 1)/2$ if x is odd and $ x/2$ if x is even. Let a be any integer with $ a \nequiv 0\; \pmod 3$. If $ {n_k}(a)$ counts the number of n with $ {T^{(k)}}(n) = a$, then for all sufficiently large k, $ {(1.302)^k} \leq {n_k}(a) \leq {(1.359)^k}$. If $ {\pi _a}(x)$ counts the number of n with $ \vert n\vert \leq x$ which eventually reach a under iteration by T, then for sufficiently large x, $ {\pi _a}(x) \geq {x^{.65}}$. The proofs are computer-intensive.

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Article copyright: © Copyright 1995 American Mathematical Society