# Unsolved Problems in Number Theory

Springer Science & Business Media, Nov 11, 2013 - Mathematics - 287 pages
To 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

 A Prime Numbers 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