Unsolved Problems in Number Theory

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 438 pages

Mathematics is kept alive by the appearance of new unsolved problems, problems posed from within mathematics itself, and also from the increasing number of disciplines where mathematics is applied. This book provides a steady supply of easily understood, if not easily solved, problems which can be considered in varying depths by mathematicians at all levels of mathematical maturity.

For this new edition, the author has included new problems on symmetric and asymmetric primes, sums of higher powers, Diophantine m-tuples, and Conway's RATS and palindromes. The author has also included a useful new feature at the end of several of the sections: lists of references to OEIS, Neil Sloane's Online Encyclopedia of Integer Sequences.

About the first Edition:

"...many talented young mathematicians will write their first papers starting out from problems found in this book." András Sárközi, MathSciNet

 

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Contents

Introduction
1
Primes of shape k 2 + 1 13 A4 The prime number race
22
A5 Arithmetic progressions of primes 25 A6 Consecutive primes
43
A16 Gaussian and EisensteinJacobi primes 55 A17 Formulas
66
B3 Unitary perfect numbers 84 B4 Amicable
91
B11 Solutions of mom no n 101 B12 Analogs with dn
105
B17 Exponentialperfect numbers 110 B18 Solutions
124
B33 Largest divisor of a binomial coefficient 134 B34 If theres an i
138
Diophantine Equations
209
D1 Sums of like powers Eulers conjecture 209 D2 The Fermat
237
D10 Exponential diophantine equations 242 D11 Egyptian
275
D20 Six general points at rational distances 287 D21 Triangles with
301
E4 Sequence with no member dividing the product of r others
314
E13 Partitioning into strongly sumfree classes 327 E14 Rados
330
with A P S 345 E24 Irrationality sequences 346 E25 Golombs
353
F1 Gaußs lattice point problem 365 F2 Lattice points with distinct
377

B42 Behavior of pan and a pn 150 B43 Alternating sums
154
C3 Lucky numbers 164 C4 Ulam numbers 166 C5 Sums
175
C11 Threesubsets with distinct sums 184 C12 The postage stamp
190
C18 The n queens problem 200 C19 Is a weakly indedendent
207
F13 Covering systems of congruences 383 F14 Exact
390
F28 A determinant of value one 401 F29 Two congruences one
403
General Index
429
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