## Unsolved Problems in Number TheoryMathematics is kept alive by the appearance of new unsolved problems, problems posed from within mathematics itself, and also from the increasing number of disciplines where mathematics is applied. This book provides a steady supply of easily understood, if not easily solved, problems which can be considered in varying depths by mathematicians at all levels of mathematical maturity. For this new edition, the author has included new problems on symmetric and asymmetric primes, sums of higher powers, Diophantine m-tuples, and Conway's RATS and palindromes. The author has also included a useful new feature at the end of several of the sections: lists of references to OEIS, Neil Sloane's Online Encyclopedia of Integer Sequences. About the first Edition: "...many talented young mathematicians will write their first papers starting out from problems found in this book." |

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### Contents

1 | |

Primes of shape k 2 + 1 13 A4 The prime number race | 22 |

A5 Arithmetic progressions of primes 25 A6 Consecutive primes | 43 |

A16 Gaussian and EisensteinJacobi primes 55 A17 Formulas | 66 |

B3 Unitary perfect numbers 84 B4 Amicable | 91 |

B11 Solutions of mom no n 101 B12 Analogs with dn | 105 |

B17 Exponentialperfect numbers 110 B18 Solutions | 124 |

B33 Largest divisor of a binomial coefficient 134 B34 If theres an i | 138 |

Diophantine Equations | 209 |

D1 Sums of like powers Eulers conjecture 209 D2 The Fermat | 237 |

D10 Exponential diophantine equations 242 D11 Egyptian | 275 |

D20 Six general points at rational distances 287 D21 Triangles with | 301 |

E4 Sequence with no member dividing the product of r others | 314 |

E13 Partitioning into strongly sumfree classes 327 E14 Rados | 330 |

with A P S 345 E24 Irrationality sequences 346 E25 Golombs | 353 |

F1 Gaußs lattice point problem 365 F2 Lattice points with distinct | 377 |

B42 Behavior of pan and a pn 150 B43 Alternating sums | 154 |

C3 Lucky numbers 164 C4 Ulam numbers 166 C5 Sums | 175 |

C11 Threesubsets with distinct sums 184 C12 The postage stamp | 190 |

C18 The n queens problem 200 C19 Is a weakly indedendent | 207 |

F13 Covering systems of congruences 383 F14 Exact | 390 |

F28 A determinant of value one 401 F29 Two congruences one | 403 |

General Index | 429 |