### Abstract

We derived from Baker’s explicit abc-conjecture that a + b = c, where a, b and c are relatively prime positive integers, implies that c < N^{1.72} for N ≥ 1 and c < 32N^{1.6} for N ≥ 1. This sharpens an estimate of Laishram and Shorey. We also show that it implies c < _{5} ^{6}N^{1+}G(N^{)} for N ≥ 3, and c < ^{6} _{5}N^{1+G1}(N^{)} for N ≥ 297856, where G(N) and G1(N) are explicitly given positive valued decreasing functions of N tending to zero as N tends to infinity. Finally, we give applications of our estimates on triples of consecutive powerful integers and generalized Fermat equation.

Original language | English |
---|---|

Pages (from-to) | 435-453 |

Number of pages | 19 |

Journal | Publicationes Mathematicae |

Volume | 94 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Abc-conjecture
- Consecutive integers
- Explicit conjecture
- Fermat’s equation

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'On Baker’s explicit abc-conjecture'. Together they form a unique fingerprint.

## Cite this

Chim, K. C., Shorey, T. N., & Sinha, S. B. (2019). On Baker’s explicit abc-conjecture.

*Publicationes Mathematicae*,*94*(3-4), 435-453. https://doi.org/10.5486/PMD.2019.8397