Solved and Unsolved Problems in Number Theory

Häftad, Engelska, 1985

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The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.

Produktinformation

Utgivningsdatum1985-02-01
FormatHäftad
SpråkEngelska
SerieAMS Chelsea Publishing
Antal sidor305
FörlagAmerican Mathematical Society
ISBN9781470476458

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Chapter I: From Perfect Numbers to the Quadratic Reciprocity Law: 1 Perfect numbers2 Euclid3 Euler's converse proved4 Euclid's algorithm5 Cataldi and others6 The prime number theorem7 Two useful theorems8 Fermat and others9 Euler's generalization proved10 Perfect numbers, II11 Euler and $M_{31}$12 Many conjectures and their interrelations13 Splitting the primes into equinumerous classes14 Euler's criterion formulated15 Euler's criterion proved16 Wilson's theorem17 Gauss's criterion18 The original Legendre symbol19 The reciprocity law20 The prime divisors of $n^2 +a$Chapter II: The Underlying Structure: 21 The residue classes as an invention22 The residue classes as a tool23 The residue classes as a group24 Quadratic residues25 Is the quadratic reciprocity law a deep theorem?26 Congruential equations with a prime modulus27 Euler's $\phi$ function28 Primitive roots with a prime modulus29 $\mathfrak{M}_{p}$ as a cyclic group30 The circular parity switch31 Primitive roots and Fermat numbers32 Artin's conjectures33 Questions concerning cycle graphs34 Answers concerning cycle graphs35 Factor generators of $\mathfrak{M}_{m}$36 Primes in some arithmetic progressions and a general divisibility theorem37 Scalar and vector indices38 The other residue classes39 The converse of Fermat's theorem40 Sufficient conditions for primalityChapter III: Pythagoreanism and Its Many Consequences: 41 The Pythagoreans42 The Pythagorean theorem43 The $\sqrt 2$ and the crisis44 The effect upon geometry45 The case for Pythagoreanism46 Three Greek problems47 Three theorems of Fermat48 Fermat's last ``Theorem''49 The easy case and infinite descent50 Gaussian integers and two applications51 Algebraic integers and Kummer's theorem52 The restricted case, Sophie Germain, and Wieferich53 Euler's ``Conjecture''54 Sum of two squares55 A generalization and geometric number theory56 A generalization and binary quadratic forms57 Some applications58 The significance of Fermat's equation59 The main theorem60 An algorithm61 Continued fractions for $\sqrt N$62 From Archimedes to Lucas63 The Lucas criterion64 A probability argument65 Fibonacci numbers and the original Lucas testAppendix to Chapters I-III: Supplementary comments, theorems, and exercisesChapter IV: Progress: 66 Chapter I fifteen years later67 Artin's conjectures, II68 Cycle graphs and related topics69 Pseudoprimes and primality70 Fermat's last ``Theorem,'' II71 Binary quadratic forms with negative discriminants72 Binary quadratic forms with positive discriminants73 Lucas and Pythagoras74 The progress report concluded75 The second progress report begins76 On judging conjectures77 On judging conjectures, II78 Subjective judgement, the creation of conjectures and inventions79 Fermat's last ``Theorem,'' III80 Computing and algorithms81 $\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)$ and all that82 1993Appendix: Statement on fundamentalsTable of definitionsReferencesIndex.