Integer Complexity: Experimental and Analytical Results II

Čerņenoks, Juris; Iraids, Janis; Opmanis, Mārtiņš; Opmanis, Rihards; Podnieks, Karlis

doi:10.48550/arxiv.1409.0446

Cited by 1 publication

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“…Guy popularized the problem in his survey [6] and also included it in his book Unsolved Problems in Number Theory [7]. Recently, there has been renewed interest [1,2,3,4,8,5,14] in the problem -the focus of our paper is on the asymptotic size of f (n).…”

Section: Introductionmentioning

confidence: 99%

“…The current best result is due to Fuller, Arias de Reyna, and Van de Lune [4] and is based on similar considerations carried out in base 3359232. Experimental results suggest that this can be improved: Iraids, Balodis, Cerņenoks, Opmanis, Opmanis, and Podnieks [5] have conjectured, based on extensive numerical computation, that lim sup n→∞ f (n) log 3 n ≤ 3.37.…”

Section: Introductionmentioning

confidence: 99%

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An Application of Markov Chain Analysis to Integer Complexity

Shriver

2015

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The complexity f (n) of an integer was introduced in 1953 by Mahler & Popken: it is defined as the smallest number of 1's needed in conjunction with arbitrarily many +, * and parentheses to write an integer n (for example, f (6) ≤ 5 since 6 = (1 + 1)(1 + 1 + 1)). The best known bounds are 3 log 3 n ≤ f (n) ≤ 3.635 log 3 n. The lower bound is due to Selfridge (with equality for powers of 3); the upper bound was recently proven by Arias de Reyna & Van de Lune, and holds on a set of natural density one.We use Markov chain methods to analyze a large class of algorithms, including one found by David Bevan that improves the upper bound to f (n) ≤ 3.52 log 3 n on a set of logarithmic density one.

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