[内容简介]
Unsolved Problems in Number Theory contains discussions of hundreds of open questions, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, not only extensive new material has been added, but corrections and additions have been included throughout the book.

[Table Of Contents]

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Glossary of Symbols

Introduction

             Prime Numbers

                   Prime values of quadratic functions

                     Primes connected with factorials

                     Mersenne primes. Repunits. Fermat numbers. Primes of shape k ? 2n + 1

                   The prime number race

                     Arithmetic progressions of primes

                     Consecutive primes in A.P.

                     Cunningham chains

                   Gaps between primes. Twin primes

                     Patterns of primes

                     Gilbreath's conjecture

                     Increasing and decreasing gaps

                     Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes

                     Carmichael numbers

                     ``Good'' primes and the prime number graph

                     Congruent products of consecutive numbers

                     Gaussian and Eisenstein-Jacobi primes

                     Formulas for primes

                   The Erdos-Selfridge classification of primes

                     Values of n making n -- 2k prime. Odd numbers not of the form ±pa ±2b

                     Symmetric and asymmetric primes

            Divisibility

                     Perfect numbers

                     Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers

                     Unitary perfect numbers

                     Amicable numbers

                     Quasi-amicable or betrothed numbers

                     Aliquot sequences

                     Aliquot cycles. Sociable numbers

                     Unitary aliquot sequences

                     Superperfect numbers

                     Untouchable numbers

                     Solutions of mσ(m) = nσ(n)

                     Analogs with d(n), σk(n)

                     Solutions of σ(n) = σ(n + 1)

                   Some irrational series

                     Solutions of σ(q) + σ(r) = σ(q + r)

                     Powerful numbers. Squarefree numbers

                     Exponential-perfect numbers

                     Solutions of d(n) = d(n + 1)

                   (m, n + 1) and (m + 1, n) with same set of prime factors. The abc-conjecture

                     Cullen and Woodall numbers

                   k ? 2n + 1 composite for all n

                     Factorial n as the product of n large factors

                   Equal products of factorials

                   The largest set with no member dividing two others

                   Equal sums of geometric progressions with prime ratios

                     Densest set with no l pairwise coprime

                   The number of prime factors of n + k which don't divide n + i, 0 ≤ i < k

                     Consecutive numbers with distinct prime factors

                   Is x determined by the prime divisors of x + 1, x + 2,..., x + k?

                   A small set whose product is square

                     Binomial coefficients

                     Grimm's conjecture

                     Largest divisor of a binomial coefficient

                   If there's an i such that n -- i divides (nk)

                     Products of consecutive numbers with the same prime factors

                     Euler's totient function

                   Does φ(n) properly divide n -- 1?

                     Solutions of φ(m) = σ(n)

                     Carmichael's conjecture

                   Gaps between totatives

                     Iterations of φ and σ

                     Behavior of φ(σ(n)) and σ(φ(n))

                     Alternating sums of factorials

                   Sums of factorials

                   Euler numbers

                   The largest prime factor of n

                   When does 2a -- 2b divide na -- nb?

                     Products taken over primes

                   Smith numbers

                     Ruth-Aaron numbers

            Additive Number Theory

                     Goldbach's conjecture

                   Sums of consecutive primes

                   Lucky numbers

                   Ulam numbers

                   Sums determining members of a set

                     Addition chains. Brauer chains. Hansen chains

                   The money-changing problem

                   Sets with distinct sums of subsets

                     Packing sums of pairs

                     Modular difference sets and error correcting codes

                     Three-subsets with distinct sums

                   The postage stamp problem

                   The corresponding modular covering problem. Harmonious labelling of graphs

                     Maximal sum-free sets

                     Maximal zero-sum-free sets

                     Nonaveraging sets. Nondividing sets

                   The minimum overlap problem

                   The n queens problem

                   Is a weakly indedendent sequence the finite union of strongly independent ones?

                   Sums of squares

                   Sums of higher powers

              Diophantine Equations

                   Sums of like powers. Euler's conjecture

                   The Fermat problem

                     Figurate numbers

                     Waring's problem. Sums of l kth Powers

                   Sum of four cubes

                   An elementary solution of x2 = 2y4 -- 1

                   Sum of consecutive powers made a power

                   A pyramidal diophantine equation

                     Catalan conjecture. Difference of two powers

                     Exponential diophantine equations

                     Egyptian fractions

                     Markoff numbers

                   The equation xx yy = zz

                   ai + bj made squares

                     Numbers whose sums in pairs make squares

                     Triples with the same sum and same product

                     Product of blocks of consecutive integers not a power

                   Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square

                     Rational distances from the corners of a square

                   Six general points at rational distances

                     Triangles with integer edges, medians and area

                     Simplexes with rational contents

                   Some quartic equations

                   Sum equals product

                     Equations involving factorial n

                     Fibonacci numbers of various shapes

                     Congruent numbers

                   A reciprocal diophantine equation

                     Diophantine m-tuples

              Sequences of Integers

                   A thin sequence with all numbers equal to a member plus a prime

                     Density of a sequence with l.c.m. of each pair less than x

                     Density of integers with two comparable divisors

                     Sequence with no member dividing the product of r others

                     Sequence with members divisible by at least one of a given set

                     Sequence with sums of pairs not members of a given sequence

                   A series and a sequence involving primes

                     Sequence with no sum of a pair a square

                     Partitioning the integers into classes with numerous sums of pairs

                     Theorem of van der Waerden. Szemeredi's theorem. Partitioning the integers into classes; at least one contains an A.P.

                     Schur's problem. Partitioning integers into sum-free classes

                   The modular version of Schur's problem

                     Partitioning into strongly sum-free classes

                     Rado's generalizations of van der Waerden's and Schur's problems

                   A recursion of Gobel

                   The 3x + 1 problem

                     Permutation sequences

                     Mahler's Z-numbers

                   Are the integer parts of the powers of a fraction infinitely often prime?

                     Davenport-Schinzel sequences

                     Thue-Morse sequences

                     Cycles and sequences containing all permutations as subsequences

                     Covering the integers with A.P.s.

                     Irrationality sequences

                     Golomb's self-histogramming sequence

                     Epstein's Put-or-Take-a-Square game

                   Max and mex sequences

                     B2-sequences. Mian-Chowla sequences

                     Sequence with sums and products all in one of two classes

                     MacMahon's prime numbers of measurement

                   Three sequences of Hofstadter

                     B2-sequences from the greedy algorithm

                     Sequences containing no monotone A.P.s.

                   Happy numbers

                   The Kimberling shuffle

                     Klarner-Rado sequences

                     Mousetrap

                   Odd sequences

            None of the Above

                     Gauß's lattice point problem

                     Lattice points with distinct distances

                     Lattice points, no four on a circle

                   The no-three-in-line problem

                     Quadratic residues. Schur's conjecture

                     Patterns of quadratic residues

                   A cubic analog of a Bhaskara equation

                     Quadratic residues whose differences are quadratic residues

                     Primitive roots

                     Residues of powers of two

                     Distribution of residues of factorials

                   How often are a number and its inverse of opposite parity?

                     Covering systems of congruences

                   Exact covering systems

                   A problem of R. L. Graham

                     Products of small prime powers dividing n

                     Series associated with the ζ-function

                   Size of the set of sums and products of a set

                     Partitions into distinct primes with maximum product

                     Continued fractions

                   All partial quotients one or two

                     Algebraic numbers with unbounded partial quotients

                   Small differences between powers of 2 and 3

                   Some decimal digital problems

                   The persistence of a number

                     Expressing numbers using just ones

                     Mahler's generalization of Farey series

                   A determinant of value one

                   Two congruences, one of which is always solvable

                   A polynomial whose sums of pairs of values are all distinct

                     Miscellaneous digital problems

                     Conway's RATS and palindromes

 Index of Authors Cited

General Index