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Description:

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… More 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." Andrs Srkzi, MathSciNetLess

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Book details

List price: $99.99 Edition: 3rd Copyright year: 2004 Publisher: Springer New York Publication date: 7/13/2004 Binding: Hardcover Pages: 438 Size: 6.10" wide x 9.25" long x 0.50" tall Weight: 1.738

AuthorTable of Contents

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

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