### Abstract

We estimate the least prime factor p of the binomial coefficient (£) for k ≥ 2. The conjecture that p ≤ max(N/k, 29) is supported by considerable numerical evidence. Call a binomial coefficient good if p > k. For 1≤ i ≤ k write N - k + i =a_{i}b_{i}where b¡ contains just those prime factors > k, and define the deficiency of a good binomial coefficient as the number of i for which b_{i}= 1. Let g(k) be the least integer N≥ k + 1 such that (£) is good. The bound g(k) ≥ ck^{2}/lnk is proved. We conjecture that our list of 17 binomial coefficients with deficiency ≥1 is complete, and it seems that the number with deficiency 1 is finite. All (^{n} _{k}) with positive deficiency and k ≤ 101 are listed.

Original language | English |
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Pages (from-to) | 215-224 |

Number of pages | 10 |

Journal | Mathematics of Computation |

Volume | 61 |

Issue number | 203 |

DOIs | |

Publication status | Published - 1993 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Mathematics of Computation*,

*61*(203), 215-224. https://doi.org/10.1090/S0025-5718-1993-1199990-6