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

Applegate, David; Lagarias, Jeffrey C.

doi:10.1090/S0025-5718-1995-1270613-2

Density bounds for the $3x+1$ problem. II. Krasikov inequalities
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by David Applegate and Jeffrey C. Lagarias PDF

Math. Comp. 64 (1995), 427-438 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 ${\pi _a}(x)$ counts the number of n with $|n| \leq x$ 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.

References

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