### Abstract

We prove that every integer n⩾10 such that n≢1 mod 4 can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author’s numerical computation regarding the generalised Riemann hypothesis to extend the explicit bounds of Ramaré–Rumely.

Original language | English |
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Pages (from-to) | 16-24 |

Number of pages | 9 |

Journal | LMS Journal of Computation and Mathematics |

Volume | 19 |

Issue number | 1 |

Early online date | 29 Jan 2016 |

DOIs | |

Publication status | Published - 1 Feb 2016 |

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### Keywords

- 11N13
- 11P32 (primary)