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Prime Numbers: The Most Mysterious Figures in Math

David Wells
John Wiley
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[Reviewed by
Judy Holdener
, on

David Wells’ Prime Numbers: The Most Mysterious Figures in Math is an A-to-Z guide to the prime numbers aimed at a general audience. To be honest, after reading the cover of the book, I was initially skeptical. Do we really need another popular book about the primes? Why can’t a potential reader use one of the myriad of resources already available (e.g., Wikipedia) to look up unfamiliar terms relating to prime numbers?

My skepticism quickly waned, however, after reading just a few pages. This “guide,” which covers a wide range of topics (well-known constants, functions and numbers relating to the primes, open problems, celebrated theorems, and famous number theorists), is not meant to be a resource at all. Wells’ A-to-Z format is merely a mechanism for introducing and explaining some of the most intriguing results and ideas in mathematics. His approach works beautifully.

First, Wells’ entries are easy to read. Wells supports formal definitions with intuitive explanations, and he provides concrete examples when possible. His discussion is lively, as he includes historical information and interesting anecdotes about the people involved. Mathematicians are interesting people. Wells is also successful in portraying the viewpoint of a mathematician as well as the culture of the mathematics community. For instance, under the entry “trivia” Wells does a good job of conveying the fact that there are certain numerical curiosities (iccanobiF primes and James Bond primes) that appeal to amateurs, but have no significance to a mathematician. Expanding on this, he then includes an example of mathematical trivia that turns out to have mathematical significance!

Like information published on-line, this book does not have to be read in a linear fashion. Wells discusses the rich connections that exist between the entries in his book, and these connections are further emphasized by the cross-references that he provides. In this way, the reader is encouraged to read about several closely related topics in one sitting. Perhaps most important, I found that any given topic in the book could be read in a five- or ten-minute sitting. Busy readers can enjoy this book in snippets, making good use of the short windows of time that are available to them.

I often tell my students that learning mathematics is like drinking good wine… it is best done in sips, so that it can be truly savored. Wells’ A-to-Z guide is, indeed, like a bottle of good wine, and the reader interested in recreational mathematics will enjoy taking sips from it.

Judy Holdener ( is an associate professor of mathematics at Kenyon College in Ohio. Her mathematical interests include number theory, algebra, and dynamical systems. As a mother of two red-headed little boys (ages 5 and 1), Judy only manages to read books “in snippets” these days.



Author’s Note.


Entries A to Z.

abc conjecture.

abundant number.

AKS algorithm for primality testing.

aliquot sequences (sociable chains).


amicable numbers.

amicable curiosities.

Andrica’s conjecture.

arithmetic progressions, of primes.

Aurifeuillian factorization.

average prime.

Bang’s theorem.

Bateman’s conjecture.

Beal’s conjecture, and prize.

Benford’s law.

Bernoulli numbers.

Bernoulli number curiosities.

Bertrand’s postulate.

Bonse’s inequality.

Brier numbers.

Brocard’s conjecture.

Brun’s constant.

Buss’s function.

Carmichael numbers.

Catalan’s conjecture.

Catalan’s Mersenne conjecture.

Champernowne’s constant.

champion numbers.

Chinese remainder theorem.

cicadas and prime periods.

circle, prime.

circular prime.

Clay prizes, the.


concatenation of primes.


consecutive integer sequence.

consecutive numbers.

consecutive primes, sums of.

Conway’s prime-producing machine.

cousin primes.

Cullen primes.

Cunningham project.

Cunningham chains.

decimals, recurring (periodic).

the period of 1/13.

cyclic numbers.

Artin’s conjecture.

the repunit connection.

magic squares.

deficient number.

deletable and truncatable primes.

Demlo numbers.

descriptive primes.

Dickson’s conjecture.

digit properties.

Diophantus (c. AD 200; d. 284).

Dirichlet’s theorem and primes in arithmetic series.

primes in polynomials.

distributed computing.

divisibility tests.

divisors (factors).

how many divisors? how big is d(n)?

record number of divisors.

curiosities of d(n).

divisors and congruences.

the sum of divisors function.

the size of σ(n).

a recursive formula.

divisors and partitions.

curiosities of σ(n).

prime factors.

divisor curiosities.

economical numbers.

Electronic Frontier Foundation.

elliptic curve primality proving.


Eratosthenes of Cyrene, the sieve of.

Erdös, Paul (1913–1996).

his collaborators and Erdös numbers.


Euclid (c. 330–270 BC).

unique factorization.

√2 is irrational.

Euclid and the infinity of primes.

consecutive composite numbers.

primes of the form 4n +3.

a recursive sequence.

Euclid and the first perfect number.

Euclidean algorithm.

Euler, Leonhard (1707–1783).

Euler’s convenient numbers.

the Basel problem.

Euler’s constant.

Euler and the reciprocals of the primes.

Euler’s totient (phi) function.

Carmichael’s totient function conjecture.

curiosities of φ(n).

Euler’s quadratic.

the Lucky Numbers of Euler.


factors of factorials.

factorial primes.

factorial sums.

factorials, double, triple . . . .

factorization, methods of.

factors of particular forms.

Fermat’s algorithm.

Legendre’s method.

congruences and factorization.

how difficult is it to factor large numbers?

quantum computation.

Feit-Thompson conjecture.

Fermat, Pierre de (1607–1665).

Fermat’s Little Theorem.

Fermat quotient.

Fermat and primes of the form x2 + y2.

Fermat’s conjecture, Fermat numbers, and Fermat primes.

Fermat factorization, from F5 to F30.

Generalized Fermat numbers.

Fermat’s Last Theorem.

the first case of Fermat’s Last Theorem.

Wall-Sun-Sun primes.

Fermat-Catalan equation and conjecture.

Fibonacci numbers.

divisibility properties.

Fibonacci curiosities.

Édouard Lucas and the Fibonacci numbers.

Fibonacci composite sequences.

formulae for primes.

Fortunate numbers and Fortune’s conjecture.

gaps between primes and composite runs.

Gauss, Johann Carl Friedrich (1777–1855).

Gauss and the distribution of primes.

Gaussian primes.

Gauss’s circle problem.

Gilbreath’s conjecture.

GIMPS—Great Internet Mersenne Prime Search.

Giuga’s conjecture.

Giuga numbers.

Goldbach’s conjecture.

good primes.

Grimm’s problem.

Hardy, G. H. (1877–1947).

Hardy-Littlewood conjectures.

heuristic reasoning.

a heuristic argument by George Pólya.

Hilbert’s 23 problems.

home prime.

hypothesis H.

illegal prime.

inconsummate number.


jumping champion.

k-tuples conjecture, prime.

knots, prime and composite.

Landau, Edmund (1877–1938).

left-truncatable prime.

Legendre, A. M. (1752–1833).

Lehmer, Derrick Norman (1867–1938).

Lehmer, Derrick Henry (1905–1991).

Linnik’s constant.

Liouville, Joseph (1809–1882).

Littlewood’s theorem.

the prime numbers race.

Lucas, Édouard (1842–1891).

the Lucas sequence.

primality testing.

Lucas’s game of calculation.

the Lucas-Lehmer test.

lucky numbers.

the number of lucky numbers and primes.

“random” primes.

magic squares.

Matijasevic and Hilbert’s 10th problem.

Mersenne numbers and Mersenne primes.

Mersenne numbers.

hunting for Mersenne primes.

the coming of electronic computers.

Mersenne prime conjectures.

the New Mersenne conjecture.

how many Mersenne primes?

Eberhart’s conjecture.

factors of Mersenne numbers.

Lucas-Lehmer test for Mersenne primes.

Mertens constant.

Mertens theorem.

Mills’ theorem.

Wright’s theorem.

mixed bag.

multiplication, fast.

Niven numbers.

odd numbers as p + 2a2.

Opperman’s conjecture.

palindromic primes.

pandigital primes.

Pascal’s triangle and the binomial coefficients.

Pascal’s triangle and Sierpinski’s gasket.

Pascal triangle curiosities.

patents on prime numbers.

Pépin’s test for Fermat numbers.

perfect numbers.

odd perfect numbers.

perfect, multiply.

permutable primes.

π, primes in the decimal expansion of.

Pocklington’s theorem.

Polignac’s conjectures.

Polignac or obstinate numbers.

powerful numbers.

primality testing.

probabilistic methods.

prime number graph.

prime number theorem and the prime counting function.


elementary proof.

record calculations.

estimating p(n).

calculating p(n).

a curiosity.

prime pretender.

primitive prime factor.

primitive roots.

Artin’s conjecture.

a curiosity.


primorial primes.

Proth’s theorem.

pseudoperfect numbers.


bases and pseudoprimes.

pseudoprimes, strong.

public key encryption.

pyramid, prime.

Pythagorean triangles, prime.

quadratic residues.

residual curiosities.

polynomial congruences.

quadratic reciprocity, law of.

Euler’s criterion.

Ramanujan, Srinivasa (1887–1920).

highly composite numbers.

randomness, of primes.

Von Sternach and a prime random walk.

record primes.

some records.

repunits, prime.

Rhonda numbers.

Riemann hypothesis.

the Farey sequence and the Riemann hypothesis.

the Riemann hypothesis and σ(n), the sum of divisors function.

squarefree and blue and red numbers.

the Mertens conjecture.

Riemann hypothesis curiosities.

Riesel number.

right-truncatable prime.

RSA algorithm.

Martin Gardner’s challenge.

RSA Factoring Challenge, the New.

Ruth-Aaron numbers.

Scherk’s conjecture.


sexy primes.

Shank’s conjecture.

Siamese primes.

Sierpinski numbers.

Sierpinski strings.

Sierpinski’s quadratic.

Sierpinski’s φ(n) conjecture.

Sloane’s On-Line Encyclopedia of Integer Sequences.

Smith numbers.

Smith brothers.

smooth numbers.

Sophie Germain primes.

safe primes.

squarefree numbers.

Stern prime.

strong law of small numbers.

triangular numbers.


twin primes.

twin curiosities.

Ulam spiral.

unitary divisors.

unitary perfect.

untouchable numbers.

weird numbers.

Wieferich primes.

Wilson’s theorem.

twin primes.

Wilson primes.

Wolstenholme’s numbers, and theorems.

more factors of Wolstenholme numbers.

Woodall primes.

zeta mysteries: the quantum connection.

Appendix A: The First 500 Primes.

Appendix B: Arithmetic Functions.