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**An introduction to the theory of numbers. 5th ed.**
*(English)*
Zbl 0423.10001

Oxford etc.: Oxford at the Clarendon Press. xvi, 426 p. hbk: £17.50; pbk: £8.00 (1979).

From the preface: The main changes in this edition are in the notion at the end of each chapter. I have sought to provide up-to-date reference for the reader who whishes to pursue a particular topic further and to present, both in the notes and in the text, a reasonably accurate account of the present state of knowledge. …There is a new, more transparent proof of Theorem 445 [on uniform distribution modulo of \((n\vartheta)\) – the reviewer] and an account of my changed opinion about Theodorus’ method in irrationals. To facilitate the use of this edition for references purposes, I have, so far as possible, kept the page numbers unchanged. For this reason, I have added a short appendix on recent progress in some aspects of the theory of prime numbers, rather than insert the material in the appropriate places in the text.

For the review of the 4th ed. see Zbl 0086.25803.

For the review of the 4th ed. see Zbl 0086.25803.

Reviewer: Andrzej Mąkowski (Warszawa)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11Axx | Elementary number theory |

11Mxx | Zeta and \(L\)-functions: analytic theory |

11Nxx | Multiplicative number theory |

### Digital Library of Mathematical Functions:

§20.12(i) Number Theory ‣ §20.12 Mathematical Applications ‣ Applications ‣ Chapter 20 Theta FunctionsChapter 27 Functions of Number Theory

### Online Encyclopedia of Integer Sequences:

Euler totient function phi(n): count numbers <= n and prime to n.From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.

Numbers that are the sum of 2 nonzero squares.

pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...

Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

Least positive primitive root of n-th prime.

Primes of the form 4*k + 3.

Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.

Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.

Primes of the form k^2 + 1.

Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.

Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).

Numbers k such that k^2 + 1 is prime.

Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

Denominators of Bernoulli numbers B_0, B_1, B_2, B_4, B_6, ...

Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

Numbers n such that sqrt(-1) mod n exists; or, numbers n that are primitively represented by x^2 + y^2.

From Euler’s Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).

Number of (positive) squarefree numbers < n.

Numbers that are not the sum of 2 primes.

Number of integer solutions to x^2 + y^2 <= n excluding (0,0).

Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power when a(n) = that prime.

a(n) = 10^n - 9^n.

Waring’s problem: least positive integer requiring maximum number of terms when expressed as a sum of positive n-th powers.

Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

a(n) = tau(n)^2, where tau(n) = A000005(n).

Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

Nearest integer to Sum_{k=1..n} log(k) = log(n!).

Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.

Sum of divisors of n that are not divisible by 4.

Nearest integer to n*log(n).

Numbers m such that the Bernoulli number B_{2*m} has denominator 30.

Numbers m such that the Bernoulli number B_m has denominator 30.

Numbers m such that the Bernoulli number B_{2*m} has denominator 42.

As p runs through the primes == 1 mod 3, sequence gives Bernoulli(2p) - 1/6.

Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r.

a(n) = Product_{i=2..n} (prime(i) - 2).

a(n) = Product_{i=3..n} (prime(i) - 3).

a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Decimal expansion of alpha(2) = Sum_{i>0} prime(i)*2^(-i^2).

Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).

As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.

Simple quadratic fields (i.e., with a unique prime factorization).

Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.

Find smallest k such that k^n is a sum of n n-th powers, say k^n = T(n,1)^n + ... + T(n,n)^n. Sequence gives triangle of successive rows T(n,1), ..., T(n,n). T(n,1) = ... = T(n,n) = 0 indicates no solution exists.

Numerators in partial products of the twin prime constant.

Denominators in partial products of the twin prime constant.

Triangle of falling factorials, read by rows: T(n, k) = n*(n-1)*...*(n-k+1), n > 0, 1 <= k <= n.

Waring’s problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.

Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.

Number of primes of the form x^2 + 1 < 10^n.

a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.

a(n) is the largest prime of the form x^2 + 1 <= 10^n.

a(n) = number of primes of the form x^2 + 1 <= 2^n.

a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 2^n.

a(n) is the largest prime of the form x^2 + 1 <= 2^n.

Smallest number expressible as the sum of three 4th powers in at least n ways.

a(n) = floor( prime(n)/log(n) ).

Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler’s constant A001620.

Moebius mu sequence for real quadratic extension sqrt(2).

a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

Decimal expansion of -x, where x is the real root of f(x) = 1 + (twin_prime(n))x^n.

a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).

(Number of squarefree numbers <= n) minus round(n/zeta(2)).

n such that the Moebius function take successively, from n, the values -1,0,-1,0,-1,0.

Smallest m such that the Moebius function takes successively, from m, n values 1,0,1,0,... ending with 1 or 0.

Numerator of product_{k=1..n-1} (1 + 1/prime(k)).

Denominator of product_{k=1..n-1} (1 + 1/prime(k)).

Values of D for which the imaginary quadratic field Q[ sqrt(-D) ] is norm-Euclidean.

Integers k such that 8*k + 1 is a prime or a square.