## Unsolved Problems in Number TheoryTo many laymen, mathematicians appear to be problem solvers, people who do "hard sums". Even inside the profession we dassify ouselves as either theorists or problem solvers. Mathematics is kept alive, much more than by the activities of either dass, by the appearance of a succession of unsolved problems, both from within mathematics itself and from the increasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solution of a problem may stifte interest in the area around it. But "Fermat 's Last Theorem", because it is not yet a theorem, has generated a great deal of "good" mathematics, whether goodness is judged by beauty, by depth or by applicability. To pose good unsolved problems is a difficult art. The balance between triviality and hopeless unsolvability is delicate. There are many simply stated problems which experts tell us are unlikely to be solved in the next generation. But we have seen the Four Color Conjecture settled, even if we don't live long enough to learn the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfect numbers. On the other hand, "unsolved" problems may not be unsolved at all, or than was at first thought. |

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### Contents

3 | |

A5 Arithmetic progressions of primes 15 A6 Consecutive primes | 26 |

A16 Gaussian primes EisensteinJacobi primes 33 A17 Formulas | 41 |

B3 Unitary perfect numbers 53 B4 Amicable | 59 |

of a q + orr a q+ r 69 B16 Powerful numbers | 70 |

B23 Equal products of factorials 79 B24 The largest set with | 84 |

B37 Does pn properly divide n 12 92 B38 Solutions | 99 |

C3 Lucky numbers 108 C4 Ulam numbers 109 C5 Sums | 115 |

D11 Egyptian fractions 158 D12 Markoff numbers 166 D13 | 169 |

differences are square 173 D19 Rational distances from the corners | 185 |

E Sequences of Integers | 199 |

E13 Partitioning into strongly sumfree classes 213 E14 Rados | 215 |

E26 Epsteins PutorTakeaSquare game 226 E27 Max and | 231 |

E33 Sequences containing no monotone A P s 233 E34 Happy | 238 |

F12 How often are a number and its inverse of opposite parity? | 251 |

F24 Squares with just two different decimal digits 262 F25 | 266 |

Harmonious labelling of graphs 127 C14 Maximal sumfree sets | 128 |

of a 29 1 152 D7 Sum of consecutive powers made a power | 153 |

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### Common terms and phrases

Acta Arith algorithm aliquot sequences amicable numbers amicable pairs arithmetic progressions asks binomial coefficients Bull Canad Carl Pomerance Carmichael numbers Colloq Comput Conf congruences conjecture consecutive integers consecutive primes covering systems cuboid D. H. Lehmer Davenport-Schinzel sequences density diophantine equation Discrete Math distinct prime divisors Egyptian fractions Elem Erdős Euler's example Fermat numbers Fibonacci Fibonacci Quart function graph Heath-Brown infinitely integers J. L. Selfridge lattice points least prime Leo Moser London Math lower bound Makowski Math Mersenne primes modulo Monthly nombres notes number of primes number of solutions Number Theory partition Paul Erdös positive integers prime factors prime numbers primitive Proc proof proved pseudoprimes quadratic residues R. L. Graham rational Recreational Math reine angew Richard Schinzel sets of integers showed shown Sierpiński smallest squarefree squares Straus subset sum-free theorem triangle Turán unitary perfect Univ values