Number Theory

Contents: 1. Number theory. 2. Identities for linear recurring sequences. 3. Half-totient tree. 4. Limit cycles of xy (mod x+y). 5. Rounding up to PI. 6. Fermat\'s last theorem for cubes. 7. Digit reversal sums leading to palindromes. 8. Discordance impedes square magic. 9. Least significant non-zero digit of n! 10. Geodesic diophantine boxes. 11. Highly Heronian ellipses. 12. How Leibniz might have anticipated Euler. 13. Odd-Greedy unit fraction expansions. 14. Four squares from three numbers. 15. Accidental melodies. 16. Some properties of the Lucas sequence. 17. On a unit fraction question of Erdos and Graham.18. The Greedy Algorithm for unit fractions. 19. Average of sigma(n)/n. 20. Lucas\'s primality test with factored N-1. 21. One in the chamber. 22. Fractions and characteristic recurrences. 23. Automendian triangles and magic squares. 24. Orthomagic square of squares. 25. Magic square of squares. 26. Anti-Carmichael pairs. 27. Coherent arrays of squares. 28. Mock-rational numbers. 29. Integer sequences related to PI. 30. Series within parallel resistance networks. 31. Pythagorean graphs. 32. On general palindromic numbers. 33. Minimizing the denominators of unit fraction expansions. 34. Perrin\'s sequence. 35. Unit fraction partitions. 36. Reflective and cyclic sets of primes. 37. Waring\'s problem. 38. Cyclic divisibility. 39. Unit fractions and Fibonacci. 40. Solving magic squares. 41. Concordant forms. 42. Numbers expressible as (aÙ2 - 1)(bÙ2 - 1). 43. Euclidean algorithm. 44. On the density of some exceptional primes. 45. Recurrence and pell equations.

"This book gives an undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples; exercises (with solutions) are integrated into the text. The first few chapters, covering divisibility, prime numbers and modular arithmetic, assume only basic school algebra, and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares; in particular, the last chapter gives a concise account of Fermat\'s last theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles." (jacket)