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
of σq + σr oq+r 69 B16 Powerful numbers 70 | 1 |
A1 Prime values of quadratic functions 4 A2 Primes connected with | 8 |
A5 Arithmetic progressions of primes 15 A6 Consecutive primes | 26 |
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a₁ Acta Arith algorithm aliquot sequences amicable numbers amicable pairs arithmetic progression Bull Canad Carl Pomerance Carmichael numbers Chen Jing-Run Cohen Colloq Comput Conf congruence consecutive primes cuboid D. H. Lehmer Davenport-Schinzel sequences density diophantine equation divisors Elem Erdős asks Erdős conjectures Euler's example Fermat numbers Fibonacci Fibonacci Quart finite formula function Gaussian primes graph infinitely integers J. L. Selfridge least prime Leech ln ln London Math lower bound Mąkowski Math Mersenne primes modulo Monthly nombres notes Notices Amer number of primes number of solutions Number Theory odd perfect number partition Paul Erdős Peter Hagis positive integers powers prime factors prime numbers primitive Proc proof proved pseudoperfect pseudoprimes quadratic rational Recreational Math reine angew residues Richard Rotkiewicz Schinzel Selfridge showed shown Sierpiński smallest squarefree squares sum-free theorem triangle Turán unitary perfect unitary perfect numbers Univ values Wagstaff Zahlen