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Mathematics of Computation

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Density bounds for the $ 3x+1$ problem. II. Krasikov inequalities

Authors: David Applegate and Jeffrey C. Lagarias
Journal: Math. Comp. 64 (1995), 427-438
MSC: Primary 11B83; Secondary 11Y99
MathSciNet review: 1270613
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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 $ {\pi _a}(x)$ counts the number of n with $ \vert n\vert \leq x$ which eventually reach a under iteration by T, then for all sufficiently large x, $ {\pi _a}(x) \geq {x^{.81}}$. The proof is based on solving nonlinear programming problems constructed using difference inequalities of Krasikov.

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Article copyright: © Copyright 1995 American Mathematical Society

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