Density bounds for the $3x+1$ problem. I. Tree-search method
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- by David Applegate and Jeffrey C. Lagarias PDF
- Math. Comp. 64 (1995), 411-426 Request permission
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 $|n| \leq x$ which eventually reach a under iteration by T, then for sufficiently large x, ${\pi _a}(x) \geq {x^{.65}}$. The proofs are computer-intensive.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 411-426
- MSC: Primary 11B83; Secondary 11Y99
- DOI: https://doi.org/10.1090/S0025-5718-1995-1270612-0
- MathSciNet review: 1270612