### Abstract

A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a_{i} + a_{j} with a_{i}, a_{i} ∈ A. If A is a basis, but no proper subset of A is a basis, then A is a minimal basis. A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis. In this paper, minimal bases consisting of square-free numbers are constructed, and also bases of square-free numbers no subset of which is minimal. Maximal nonbases of square-free numbers do not exist. However, nonbases of square-free numbers that are maximal with respect to the set of square-free numbers are constructed, and also nonbases of square-free numbers that are not contained in any nonbasis of square-free numbers maximal with respect to the square-free numbers.

Original language | English |
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Pages (from-to) | 197-208 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 11 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1979 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*11*(2), 197-208. https://doi.org/10.1016/0022-314X(79)90039-8