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Mathematics - Number Theory and Discrete Mathematics | Unsolved Problems in Number Theory

Unsolved Problems in Number Theory

Guy, Richard

2nd ed. 1994, XVI, 287 p.

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To many laymen, mathematicians appear to be problem solvers, people who do "hard sums". Even inside the profession we dassify ouselves as either theorists or problem solvers. Mathematics is kept alive, much more than by the activities of either dass, by the appearance of a succession of unsolved problems, both from within mathematics itself and from the increasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solution of a problem may stifte interest in the area around it. But "Fermat 's Last Theorem", because it is not yet a theorem, has generated a great deal of "good" mathematics, whether goodness is judged by beauty, by depth or by applicability. To pose good unsolved problems is a difficult art. The balance between triviality and hopeless unsolvability is delicate. There are many simply stated problems which experts tell us are unlikely to be solved in the next generation. But we have seen the Four Color Conjecture settled, even if we don't live long enough to learn the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfect numbers. On the other hand, "unsolved" problems may not be unsolved at all, or than was at first thought.

Content Level » Professional/practitioner

Keywords » Zahlentheorie

Related subjects » Number Theory and Discrete Mathematics

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 Erd½os-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 mó(m) = nó(n). B12. Analogs with d(n), ók(n). B13. Solutions of ó(n) = ó(n + 1). B14. Some irrational series. B15. Solutions of ó(q) + ó(r) = ó(q + r). B16. Powerful numbers. Squarefree 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 theproduct 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 ¡Ü i < k. 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 ö(n) properly divide n - 1? B38. Solutions of ö(m) = ó(n). B39. Carmichael's conjecture. B40. Gaps between totatives. B41. Iterations of ö and ó. B42. Behavior of ö(ó(n)) and ó(ö(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

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