## Abstract

If p is prime, then let φ_{p} denote the Legendre symbol modulo p and let ε_{p} be the trivial character modulo p. As usual, let _{n+1}F_{n}(x)_{p} := _{n+1}F_{n}(^{φp, φp, ..., φp} _{εp, ..., εp} | x )_{p} be the Gaussian hypergeometric series over double-struck F_{p}. For n > 2 the non-trivial values of _{n+1}F_{n}(x)_{p} have been difficult to obtain. Here we take the first step by obtaining a simple formula for _{4}F_{3}(1)_{p}. As a corollary we obtain a result describing the distribution of traces of Frobenius for certain families of elliptic curves. We also find that _{4}F_{3}(1)_{p} satisfies surprising congruences modulo 32 and 11. We then establish a mod p^{2} "supercongruence" between Apéry numbers and the coefficients of a certain eta-product; this relationship was conjectured by Beukers in 1987. Finally, we obtain many new mod p congruences for generalized Apéry numbers.

Original language | English (US) |
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Pages (from-to) | 187-212 |

Number of pages | 26 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 518 |

DOIs | |

State | Published - 2000 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics