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.

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 Erd¿os-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 Woodallnumbers. 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 ¡Ü i