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. |
Contents
A1 Prime values of quadratic functions 4 A2 Primes connected with | 8 |
A5 Arithmetic progressions of primes 15 A6 Consecutive primes | 26 |
A16 Gaussian primes EisensteinJacobi primes 33 A17 Formulas | 41 |
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a₁ Acta Arith algorithm aliquot sequences amicable numbers amicable pairs arithmetic progressions binomial coefficients Bull Canad Carl Pomerance Carmichael numbers Colloq Comput Conf congruences 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 asks Erdős conjectures Euler's example Fermat numbers Fibonacci Fibonacci Quart function graph Heath-Brown infinitely integers J. L. Selfridge lattice points least prime ln ln London Math lower bound Mąkowski Math Mersenne primes modulo Monthly nombres notes number of primes number of solutions Number Theory partition Paul Erdős positive integers prime factors prime numbers Proc proof proved pseudoprimes quadratic residues R. L. Graham rational reine angew Richard Schinzel sets of integers showed shown Sierpiński smallest squarefree squares Straus subset sum-free theorem triangle Turán unitary perfect Univ values