Title:
Unsolved problems in number theory
Personal Author:
Series:
Problem books in mathematics
Edition:
3rd ed.
Publication Information:
New York, NY : Springer, 2004
ISBN:
9780387208602
Subject Term:
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010138835 | QA241 G89 2004 | Open Access Book | Book | Searching... |
Searching... | 30000010156567 | QA241 G89 2004 | Open Access Book | Book | Searching... |
On Order
Summary
Summary
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
Table of Contents
| Preface to the First Edition |
| Preface to the Second Edition |
| Preface to the Third Edition |
| Glossary of Symbols |
| A Prime Numbers |
| A1 Prime values of quadratic functions |
| A2 Primes connected with factorials |
| A3 Mersenne primes |
| Repunits |
| Fermat numbers |
| Primes of shape k . 2n + 1 |
| A4 The prime number race |
| A5 Arithmetic progressions of primes |
| A6 Consecutive primes in A.P |
| A7 Cunningham chains |
| A8 Gaps between primes |
| Twin primes |
| A9 Patterns of primes |
| A10 Gilbreath's conjecture |
| A11 Increasing and decreasing gaps |
| A12 Pseudoprimes |
| Euler pseudoprimes |
| Strong pseudoprimes |
| A13 Carmichael numbers |
| A14 "Good" primes and the prime number graph |
| A15 Congruent products of consecutive numbers |
| A16 Gaussian primes |
| Eisenstein-Jacobi primes |
| A17 Formulas for primes |
| A18 The Erd1/4os-Selfridge classi.cation of primes |
| A19 Values of n making n - 2k prime |
| Odd numbers not of the form pa 2b |
| A20 Symmetric and asymmetric primes |
| B Divisibility |
| B1 Perfect numbers |
| B2 Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers |
| B3 Unitary perfect numbers |
| B4 Amicable numbers |
| B5 Quasi-amicable or betrothed numbers |
| B6 Aliquot sequences |
| B7 Aliquot cycles |
| Sociable numbers |
| B8 Unitary aliquot sequences |
| B9 Superperfect numbers |
| B10 Untouchable numbers |
| B11 Solutions of mo(m) = no(n) |
| B12 Analogs with d(n), ok(n) |
| B13 Solutions of o(n) = o(n + 1) |
| B14 Some irrational series |
| B15 Solutions of o(q) + o(r) = o(q + r) |
| B16 Powerful numbers |
| B17 Exponential-perfect numbers |
| B18 Solutions of d(n) = d(n + 1) |
| B19 (m, n + 1) and (m+1, n) with same set of prime factors |
| The abc-conjecture |
| B20 Cullen and Woodall numbers |
| B21 k . 2n + 1 composite for all n |
| B22 Factorial n as the product of n large factors |
| B23 Equal products of factorials |
| B24 The largest set with no member dividing two others |
| B25 Equal sums of geometric progressions with prime ratios |
| B26 Densest set with no l pairwise coprime |
| B27 The number of prime factors of n + k which don''t divide n + i, 0 U i |
| B28 Consecutive numbers with distinct prime factors |
| B29 Is x determined by the prime divisors of x + 1, x + 2,. . ., x + k? |
| B30 A small set whose product is square |
| B31 Binomial coeffcients |
| B32 Grimm's conjecture |
| B33 Largest divisor of a binomial coeffcient |
| B34 If there's an i such that n - i divides _nk_ |
| B35 Products of consecutive numbers with the same prime factors |
| B36 Euler's totient function |
| B37 Does o(n) properly divide n - 1? |
| B38 Solutions of o(m) = o(n) |
| B39 Carmichael's conjecture |
| B40 Gaps between totatives |
| B41 Iterations of o and o |
| B42 Behavior of o(o(n)) and o(o(n)) |
| B43 Alternating sums of factorials |
| B44 Sums of factorials |
| B45 Euler numbers |
| B46 The largest prime factor of n |
| B47 When does 2a -2b divide na - nb? |
| B48 Products taken over primes |
| B49 Smith numbers |
| C Additive Number Theory |
| C1 Goldbach's conjecture |
| C2 Sums of consecutive primes |
| C3 Lucky numbers |
| C4 Ulam numbers |
| C5 Sums determining members of a set |
| C6 Addition chains |
| Brauer chains |
| Hansen chains |
| C7 The money-changing problem |
| C8 Sets with distinct sums of subsets |
| C9 Packing sums of pairs |
| C10 Modular di.erence sets and error correcting codes |
| C11 Three-subsets with distinct sums |
| C12 The postage stamp problem |
| C13 The corresponding modular covering problem |
| Harmonious labelling of graphs |
| C14 Maximal sum-free sets |
| C15 Maximal zero-sum-free sets |
| C16 Nonaveraging sets |
| Nondividing sets |
| C17 The minimum overlap problem |
| C18 The n queens problem |
| C19 Is a weakly indedendent sequence the .nite union of strongly independent ones? |
| C20 Sums of squares |
| C21 Sums of higher powers |
| D Diophantine Equations |
| D1 Sums of like powers |
| Euler's conjecture |
| D2 The Fermat problem |
| D3 Figurate numbers |
| D4 Waring's problem |
| Sums of l kth Powers |
| D5 Sum of four cubes |
| D6 An elementary solution of x2 = 2y4 1 |
| D7 Sum of consecutive powers made a power |
| D8 A p |
