# A short note on integer complexity

@article{Steinerberger2014ASN,
title={A short note on integer complexity},
author={Stefan Steinerberger},
journal={Contributions Discret. Math.},
year={2014},
volume={9}
}
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.

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## References

SHOWING 1-10 OF 13 REFERENCES

### Numbers with Integer Complexity Close to the Lower Bound

• Mathematics
Integers
• 2012
This paper presents a method for classifying all n with for a given r, and proves that for with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m.

### Integer Complexity: Breaking the θ(n2) barrier

• Mathematics, Computer Science
• 2008
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

• Mathematics
• 2012
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.

### Complexity of the natural numbers. (Spanish)

• Gac. R. Soc. Mat. Esp
• 2000

### Complexity of the natural numbers. (Spanish)

• Gac. R. Soc. Mat. Esp
• 2000

### Unsolved problems: Some suspiciously simple sequences

• Amer. Math. Monthly
• 1986

• Wiskd
• 1953

### Interger complexity: Experimental and analytical results

• Computer Science and Information Technologies
• 2012