(F10 Residues of powers of two)

Graham asked about the residue of 2^{n} mod n.

Lehmer found the smallest solution for 2^{n}≡3 (mod n),

n=4700063497

I tried to find all the solutions of a^{n}≡c (mod n) where

a : prime under 100, 1 ≤ c ≤ 100, n ≤ 10^{9}.

The program is very simple so I just show the results of computation.

The solutions for a=3, c=14, 34, 56 and a=7, c=36 are listed in

The On-Line Encyclopedia of Integer Sequences.

a | c : no solution under 10^{10} |
---|---|

2 | 69 |

3 | 14, 34, 56, 74 |

5 | 58 |

7 | 36, 66, 86 |

11 | 52, 94 |

13 | 36, 54, 59, 80, 92 |

17 | 16, 30, 64, 100 |

19 | 62, 78, 86 |

23 | 42, 64, 84 |

29 | 18 |

31 | 24, 54, 98 |

37 | |

41 | 42 |

43 | |

47 | 36, 46, 64, 90 |

53 | 54, 64, 84, 96 |

59 | 36, 42 |

61 | 32, 78 |

67 | 50 |

71 | 88 |

73 | 78 |

79 | |

83 | 54 |

89 | 82 |

97 | 50 |

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E-mail : kc2h-msm@asahi-net.or.jpHisanori Mishima