Krasikov, Ilia; Lagarias, Jeffrey C. Bounds for the \(3x+1\) problem using difference inequalities. (English) Zbl 1069.11011 Acta Arith. 109, No. 3, 237-258 (2003). The “Collatz”-problem (or “3x+1”- or “Hasse”- or “Syracuse”- or “Kakutani”- problem) is to prove that for every \(n \in \mathbb N\) there exists a \(k\) with \(T^{(k)}(n)=1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\). Let \(\pi_1(x)\) count the number of integers below \(x\) that eventually reach \(1\) under this iteration. There are a number of estimates of the form \(\pi_1(x)>x^\gamma\) for a positive constant \(\gamma\) and sufficiently large \(x\). In the note under review the authors improve \(\gamma=0.81\), found by D. Applegate and J. C. Lagarias [see Math. Comput. 64, 411–426, 427–438 (1995; Zbl 0820.11006)], to \(\gamma=0.84\). They are able to establish that the nonlinear lower bounds from the difference inequalities introduced by the first author [see Int. J. Math. Math. Sci. 12, No. 4, 791–796 (1989; Zbl 0685.10008)] do give lower bounds for the \(3x+1\) function. Their main result uses a refined linear program family and relies on heavy computations. Reviewer: Helmut Müller (Hamburg) Cited in 6 Documents MathOverflow Questions: Heuristic for a density conjecture related to the Collatz \((3x+1)\)-problem MSC: 11B83 Special sequences and polynomials 26A18 Iteration of real functions in one variable 11B37 Recurrences Keywords:\(3x+1\) problem; difference equations; Collatz problem Citations:Zbl 0820.11006; Zbl 0685.10008 PDFBibTeX XMLCite \textit{I. Krasikov} and \textit{J. C. Lagarias}, Acta Arith. 109, No. 3, 237--258 (2003; Zbl 1069.11011) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: The Collatz or 3x+1 map: a(n) = n/2 if n is even, 3n + 1 if n is odd.