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Solved and Unsolved Problems in Number Theory

Daniel Shanks
American Mathematical Society/Chelsea
Publication Date: 
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book is a very idiosyncratic introductory text in number theory. The author’s starting point is the statement, “Much of elementary number theory arose out of the investigation of three problems; that of perfect numbers, that of periodic decimals, and that of Pythagorean numbers. We have accordingly organized the book into three long chapters.” (p. xi) The book expands and weaves together the ideas arising in these three areas to give a fairly comprehensive coverage of elementary number theory. Each problem leads to more problems, some solved and some still unsolved. It is not a problem book, but a book that uses problems to drive the exposition.

This book is now in its fourth edition (the first was in 1962, the fourth in 1993), and another idiosyncratic feature is that it was updated not by revising the text but by adding a series of supplements; these now make up one-third of the book. The supplements deal with the same problems as in the main work, but usually in more depth, and they cover what were (at time of publication) some of the latest results. My favorite part is the long discussion on the nature of conjectures beginning on p. 239. Shanks thought there should be very strong evidence in favor of a statement before it is dignified by the name “conjecture”. This book is also the source of his famous statement (regarding experimental evidence for the non-existence of odd perfect numbers), “1050 is a long way from infinity”. (p. 217)

Overall I think this approach didn’t work well for an introductory text; the idea is good, but the execution jumps around too much and will bewilder the beginning student. That said, for the professional mathematician it is fascinating to see the interconnections in these many topics. A good introductory text where consideration of particular problems and numerical evidence drive the exposition is R. P. Burn’s A Pathway Into Number Theory.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

Chapter I: From Perfect Numbers to the Quadratic Reciprocity Law

  • 1 Perfect numbers
  • 2 Euclid
  • 3 Euler's converse proved
  • 4 Euclid's algorithm
  • 5 Cataldi and others
  • 6 The prime number theorem
  • 7 Two useful theorems
  • 8 Fermat and others
  • 9 Euler's generalization proved
  • 10 Perfect numbers, II
  • 11 Euler and $M_{31}$
  • 12 Many conjectures and their interrelations
  • 13 Splitting the primes into equinumerous classes
  • 14 Euler's criterion formulated
  • 15 Euler's criterion proved
  • 16 Wilson's theorem
  • 17 Gauss's criterion
  • 18 The original Legendre symbol
  • 19 The reciprocity law
  • 20 The prime divisors of $n^2 +a$

Chapter II: The Underlying Structure

  • 21 The residue classes as an invention
  • 22 The residue classes as a tool
  • 23 The residue classes as a group
  • 24 Quadratic residues
  • 25 Is the quadratic reciprocity law a deep theorem?
  • 26 Congruential equations with a prime modulus
  • 27 Euler's $\phi$ function
  • 28 Primitive roots with a prime modulus
  • 29 $\mathfrak{M}_{p}$ as a cyclic group
  • 30 The circular parity switch
  • 31 Primitive roots and Fermat numbers
  • 32 Artin's conjectures
  • 33 Questions concerning cycle graphs
  • 34 Answers concerning cycle graphs
  • 35 Factor generators of $\mathfrak{M}_{m}$
  • 36 Primes in some arithmetic progressions and a general divisibility theorem
  • 37 Scalar and vector indices
  • 38 The other residue classes
  • 39 The converse of Fermat's theorem
  • 40 Sufficient conditions for primality

Chapter III: Pythagoreanism and Its Many Consequences

  • 41 The Pythagoreans
  • 42 The Pythagorean theorem
  • 43 The $\sqrt 2$ and the crisis
  • 44 The effect upon geometry
  • 45 The case for Pythagoreanism
  • 46 Three Greek problems
  • 47 Three theorems of Fermat
  • 48 Fermat's last "Theorem"
  • 49 The easy case and infinite descent
  • 50 Gaussian integers and two applications
  • 51 Algebraic integers and Kummer's theorem
  • 52 The restricted case, Sophie Germain, and Wieferich
  • 53 Euler's "Conjecture"
  • 54 Sum of two squares
  • 55 A generalization and geometric number theory
  • 56 A generalization and binary quadratic forms
  • 57 Some applications
  • 58 The significance of Fermat's equation
  • 59 The main theorem
  • 60 An algorithm
  • 61 Continued fractions for $\sqrt N$
  • 62 From Archimedes to Lucas
  • 63 The Lucas criterion
  • 64 A probability argument
  • 65 Fibonacci numbers and the original Lucas test

Appendix to Chapters I-III

  • Supplementary comments, theorems, and exercises

Chapter IV: Progress

  • 66 Chapter I fifteen years later
  • 67 Artin's conjectures, II
  • 68 Cycle graphs and related topics
  • 69 Pseudoprimes and primality
  • 70 Fermat's last "Theorem," II
  • 71 Binary quadratic forms with negative discriminants
  • 72 Binary quadratic forms with positive discriminants
  • 73 Lucas and Pythagoras
  • 74 The progress report concluded
  • 75 The second progress report begins
  • 76 On judging conjectures
  • 77 On judging conjectures, II
  • 78 Subjective judgement, the creation of conjectures and inventions
  • 79 Fermat's last "Theorem," III
  • 80 Computing and algorithms
  • 81 $\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)$ and all that
  • 82 1993


  • Statement on fundamentals
  • Table of definitions
  • References
  • Index