Description
Springer Unsolved Problems in Number Theory by Richard Guy
Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane's Online Encyclopedia of Integer Sequences, at the end of several of the sections._x000D_ Table of contents :- _x000D_
Preface to the First Edition_x000D_
Preface to the Second Edition_x000D_
Preface to the Third Edition_x000D_
Glossary of Symbols_x000D_
A. Prime Numbers._x000D_
A1. Prime values of quadratic functions._x000D_
A2. Primes connected with factorials._x000D_
A3. Mersenne primes. Repunits. Fermat numbers. Primes of shape k * 2n + 1. _x000D_
A4. The prime number race._x000D_
A5. Arithmetic progressions of primes._x000D_
A6. Consecutive primes in A.P. _x000D_
A7. Cunningham chains._x000D_
A8. Gaps between primes. Twin primes. _x000D_
A9. Patterns of primes._x000D_
A10. Gilbreath's conjecture._x000D_
A11. Increasing and decreasing gaps._x000D_
A12. Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes. _x000D_
A13. Carmichael numbers._x000D_
A14. 'Good' primes and the prime number graph._x000D_
A15. Congruent products of consecutive numbers. _x000D_
A16. Gaussian primes. Eisenstein-Jacobi primes. _x000D_
A17. Formulas for primes. _x000D_
A18. The Erd1/2os-Selfridge classi.cation of primes. _x000D_
A19. Values of n making n - 2k prime. Odd numbers not of the form +/-pa +/- 2b. _x000D_
A20. Symmetric and asymmetric primes. B. Divisibility _x000D_
B1. Perfect numbers. _x000D_
B2. Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers. _x000D_
B3. Unitary perfect numbers. _x000D_
B4. Amicable numbers. _x000D_
B5. Quasi-amicable or betrothed numbers. _x000D_
B6. Aliquot sequences. _x000D_
B7. Aliquot cycles. Sociable numbers. _x000D_
B8. Unitary aliquot sequences. _x000D_
B9. Superperfect numbers. _x000D_
B10. Untouchable numbers. _x000D_
B11. Solutions of mo(m) = no(n). _x000D_
B12. Analogs with d(n), ok(n). _x000D_
B13. Solutions of o(n) = o(n + 1). _x000D_
B14. Some irrational series. _x000D_
B15. Solutions of o(q) + o(r) = o(q + r). _x000D_
B16. Powerful numbers. Squarefree numbers. _x000D_
B17. Exponential-perfect numbers _x000D_
B18. Solutions of d(n) = d(n + 1). _x000D_
B19. (m, n + 1) and (m+1, n) with same set of prime factors. The abc-conjecture. B20. Cullen and Woodallnumbers. _x000D_
B21. k * 2n + 1 composite for all n. _x000D_
B22. Factorial n as the product of n large factors. _x000D_
B23. Equal products of factorials. _x000D_
B24. The largest set with no member dividing two others. _x000D_
B25. Equal sums of geometric progressions with prime ratios. _x000D_
B26. Densest set with no l pairwise coprime. _x000D_
B27. The number of prime factors of n + k which don't divide n + i, 0 !UE i < k._x000D_
B28. Consecutive numbers with distinct prime factors. _x000D_
B29. Is x determined by the prime divisors of x + 1, x + 2,. . ., x + k? _x000D_
B30. A small set whose product is square. _x000D_
B31. Binomial coeffcients. _x000D_
B32. Grimm's conjecture. _x000D_
B33. Largest divisor of a binomial coeffcient. _x000D_
B34. If there's an i such that n - i divides _nk_. _x000D_
B35. Products of consecutive numbers with the same prime factors. _x000D_
B36. Euler's totient function. _x000D_
B37. Does oe(n) properly divide n - 1? _x000D_
B38. Solutions of oe(m) = o(n). _x000D_
B39. Carmichael's conjecture. _x000D_
B40. Gaps between totatives. _x000D_
B41. Iterations of oe and o. _x000D_
B42. Behavior of oe(o(n)) and o(oe(n)). _x000D_
B43. Alternating sums of factorials. _x000D_
B44. Sums of factorials. _x000D_
B45. Euler numbers. _x000D_
B46. The largest prime factor of n. _x000D_
B47. When does 2a -2b divide na - nb? _x000D_
B48. Products taken over primes. _x000D_
B49. Smith numbers. C. Additive Number Theory _x000D_
C1. Goldbach's conjecture. _x000D_
C2. Sums of consecutive primes. _x000D_
C3. Lucky numbers. _x000D_
C4. Ulam numbers. _x000D_
C5. Sums determining members of a set. _x000D_
C6. Addition chains. Brauer chains. Hansen chains. _x000D_
C7. The money-changing problem. _x000D_
C8. Sets with distinct sums of subsets. _x000D_
C9. Packing sums of pairs. _x000D_
C10. Modular di.erence sets and error correcting codes. _x000D_
C11. Three-subsets with distinct sums. _x000D_
C12. The postage stamp problem. _x000D_
C13. The corresponding modular covering problem. Harmonious labelling of graphs. C14._x000D_