A short note on integer complexity

  title={A short note on integer complexity},
  author={Stefan Steinerberger},
  journal={Contributions Discret. Math.},
Building on an earlier approach by Isbell and Guy, this short note gives a new, constructive upper bound on the smallest number of 1's needed in conjunction with arbitrarily many +, *, and parentheses to write an integer n for generic integers n. 

An Application of Markov Chain Analysis to Integer Complexity

Markov chain methods are used to analyze a large class of algorithms, including one found by David Bevan that improves the upper bound to f(n) \leq 3.52 \log_{3}{n}$$ on a set of logarithmic density one.

On Algorithms to Calculate Integer Complexity

A method is discussed that provides a strong uniform bound on the number of summands that must be calculated for almost all $n$ and potential improvements to this algorithm are discussed.

Algorithms for determining integer complexity

Three algorithms to compute the complexity of all natural numbers, each superior to the one in [11], are presented and it is shown that they run in time $O(N^\alpha)$ and space $N\log\log N)$.

Integer Complexity: Experimental and Analytical Results II

This work considers representing of natural numbers by expressions using 1's, addition, multiplication and parentheses, and the so-called $P$-algorithms - a family of "deterministic" algorithms for building representations of numbers.



Numbers with Integer Complexity Close to the Lower Bound

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Integer Complexity: Breaking the θ(n2) barrier

An algorithm with Θ(n log 2 3) as its running time is presented and a proof of the theorem is presented: the largest solutions of f (m) = 3k, 3k±1 are, respectively, m = 3 k , 3 k ± 3 k−1.

Integer Complexity: Experimental and Analytical Results

We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the

Unsolved problems in number theory

The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.

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