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Unsolved Problems in Number Theory : Problem Books in Mathematics - Richard Guy

Unsolved Problems in Number Theory

Problem Books in Mathematics

Hardcover Published: 13th July 2004
ISBN: 9780387208602
Number Of Pages: 438

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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." Andras Sarkozi, MathSciNet

From the reviews of the third edition:

"This is the third edition of Richard Guy's well-known problem book on number theory ... . The earlier editions have served well in providing beginners as well as seasoned researchers in number theory with a good supply of problems. ... many of the problems from earlier editions have been expanded with more up-to-date comments and remarks. ... There is little doubt that a new generation of talented young mathematicians will make very good use of this book ... ." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005)

"The earlier editions of this book are among the most-opened books on the shelves of many practicing number theorists. The descriptions of state-of-the-art results on every topic and the extensive bibliographies in each section provide valuable ports of entry to the vast literature. A new and promising addition to this third edition is the inclusion of frequent references to entries in the Online encyclopedia of integer sequences at the end of each topic." (Greg Martin, Mathematical Reviews, Issue 2005 h)

Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
Glossary of Symbols
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 primes
Eisenstein-Jacobi primes
Formulas for primes
The Erd1/4os-Selfridge classi.cation 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 mo(m) = no(n)
Analogs with d(n), ok(n)
Solutions of o(n) = o(n + 1)
Some irrational series
Solutions of o(q) + o(r) = o(q + r)
Powerful 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 U 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 coeffcients
Grimm's conjecture
Largest divisor of a binomial coeffcient
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 o(n) properly divide n - 1?
Solutions of o(m) = o(n)
Carmichael's conjecture
Gaps between totatives
Iterations of o and o
Behavior of o(o(n)) and o(o(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
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 di.erence 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 .nite union of strongly independent ones?
Sums of squares
Sums of higher powers
Diophantine Equations
Sums of like powers
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 p
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387208602
ISBN-10: 0387208607
Series: Problem Books in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 438
Published: 13th July 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.27
Weight (kg): 1.81
Edition Number: 3
Edition Type: Revised

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