## Unsolved Problems in Number TheoryTo many laymen, mathematicians appear to be problem solvers, people who do "hard sums". Even inside the profession we dassify ourselves 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 in creasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solu tion 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 byapplicability. 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 ifwe don't live long enough to leam the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfeet numbers. On the other hand, "unsolved" problems may not be unsolved at all, or may be much more tractable than was at first thought. |

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### Contents

Introduction | 1 |

B Divisibility | 25 |

Additive Number Theory | 58 |

Some Diophantine Equations | 79 |

E Sequences of Integers | 110 |

F None of the Above | 132 |

Index of Authors Cited | 149 |

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### Common terms and phrases

Acta Arith aliquot sequences amicable numbers arithmetic progressions Bull Canad Carl Pomerance Carmichael numbers Colloq Comput Conf Congressus Numerantium congruences consecutive integers consecutive primes cuboid D. H. Lehmer density diophantine equation distinct prime Egyptian fractions Erdös asks Erdös conjectures Euler's example Fermat numbers Fibonacci finite number Gaussian primes H. C. Williams Hagis infinity integers J. L. Selfridge L. E. Dickson lattice points least Leo Moser London Math Makowski Mersenne primes Monthly NJ NJ nombres Notices Amer number of primes number of solutions odd perfect numbers partitioned Paul Erdös positive integers primality prime factors prime numbers Problems and results problems in number Proc proved pseudoprimes quadratic R. L. Graham reine angew residues Richard Rotkiewicz Schinzel Selfridge sequences of integers sets of integers shown Sierpiński squarefree squares subset sum-free theorem Turán twin primes unitary unitary perfect Univ unsolved problems values Zahlen