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An introduction to the theory of numbers. 3rd ed. (English) Zbl 0058.03301

Oxford: At the Clarendon Press. xvi, 419 pp. (1954).
Die dritte Auflage dieses bereits zum Standardwerk gewordenen Buches – von dem eine deutsche Übersetzung in Vorbereitung ist (München: R. Oldenbourg Verlag 1958; Zbl 0078.03101) – unterscheidet sich von den beiden vorangehenden Auflagen [(1938; Zbl 0020.29201; JFM 64.0093.03); 2nd ed. (1945)] vor allem darin, daß der inzwischen entdeckte elementare Beweis des Primzahlsatzes aufgenommen wurde, während die ersten beiden Auflagen keinen Beweis dieses Satzes brachten. Der vorgetragene Beweis beruht auf der Formel von A. SeIberg und benutzt dessen Darstellung [Ann. Math. (2) 50, 305–313 (1949; Zbl 0036.30604)], kürzt sie aber durch Verwendung von Integralen ab (vgl. E. M. Wright, Proc. R. Soc. Edinb., Sect. A 63, 257–267 (1952; Zbl 0049.16402)]).
Im übrigen sind nur an wenigen Stellen Änderungen oder Ergänzungen vorgenommen worden, z. B. Irrationalität von \(\pi\) und \(\pi^2\), Umkehrung des kleinen Fermatschen Satzes, Kriterien für Primzahleigenschaft, Einführung algebraischer Zahlen, neuere Ergebnisse in der Geometrie der Zahlen. Ebenso sind die für das Buch so charakteristischen und wertvollen Anmerkungen zu den einzelnen Kapiteln auf den neuesten Stand gebracht worden.
Die dritte Auflage mußte nach dem Tode Hardys vom zweiten der Verff. allein besorgt werden. Man darf ihm dafür danken, daß er Stil und Eigenart des Buches so lebendig erhalten hat.


Inhalt (vom Referat zur 1. Aufl. übernommen):
1. Primzahlen (1. Teil). 2. Primzahlen (2. Teil). 3. Fareyreihen und ein Satz von Minkowski (über Punktgitter). 4. Irrationalzahlen. 5. Kongruenzen und Reste (Grundbegriffe). 6. Fermatscher Satz und Folgerungen. 7. Allgemeine Eigenschaften der Kongruenzen. 8. Kongruenzen nach zusammengesetzten Moduln. 9. Darstellung von Zahlen in Ziffernsystemen. 10. Kettenbrüche. 11. Approximation von irrationalen Zahlen durch rationale. 12. Der Fundamentalsatz der Zahlentheorie in den Körpern \(k(l)\), \(k(i)\) und \(k(\rho)\). 13. Einige diophantische Gleichungen. 14. Quadratische Zahlkörper (1. Teil). 15. Quadratische Zahlkörper (2. Teil). 16. Die zahlentheoretischen Funktionen \(\varphi(n)\), \(\mu(n)\), \(d(n)\), \(\sigma(n)\), \(r(n)\). 17. Erzeugende Funktionen für zahlentheoretische Funktionen. 18. Die Größenordnung von zahlentheoretischen Funktionen. 19. Partitionen. 20. Darstellung von Zahlen durch zwei oder vier Quadrate. 21. Darstellung durch Kuben und höhere Potenzen. 22. Primzahlen (3. Teil). 23. Kroneckers Approximationssatz. 24. Einige weitere Sätze von Minkowski (über Linearformen).

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11Axx Elementary number theory
11Hxx Geometry of numbers
11Jxx Diophantine approximation, transcendental number theory
11Mxx Zeta and \(L\)-functions: analytic theory
11Nxx Multiplicative number theory
11Pxx Additive number theory; partitions
11Rxx Algebraic number theory: global fields

Online Encyclopedia of Integer Sequences:

Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.
Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.
The prime numbers.
Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.
Number of ways of writing n as a sum of 6 squares.
Number of ways of writing n as a sum of 8 squares.
Number of ways of writing n as a sum of 10 squares.
Number of ways of writing n as a sum of 12 squares.
Number of ways of writing n as a sum of 16 squares.
Number of ways of writing n as a sum of 24 squares.
Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
Fermat numbers: a(n) = 2^(2^n) + 1.
Pentagonal numbers: a(n) = n*(3*n-1)/2.
Sums of three squares: numbers of the form x^2 + y^2 + z^2.
Perfect numbers k: k is equal to the sum of the proper divisors of k.
Numbers that are the sum of 3 but no fewer nonzero squares.
Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes.
Numerators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.
Mersenne numbers: 2^p - 1, where p is prime.
a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.
Number of partitions of n into at most 4 parts.
a(0) = 3; thereafter, a(n) = a(n-1)^2 - 2.
Full reptend primes: primes with primitive root 10.
(Presumed) solution to Waring’s problem: g(n) = 2^n + floor((3/2)^n) - 2.
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
Positive integers D such that Q[sqrt(D)] is a quadratic field which is norm-Euclidean.
Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).
Triangle read by rows: row n gives numerators of Farey series of order n.
Triangle read by rows: row n gives denominators of Farey series of order n.
Long period primes: the decimal expansion of 1/p has period p-1.
Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.
Numerators of Farey (or Stern-Brocot) tree fractions.
a(n) = mu(n) + 1, where mu is the Moebius function.
Smallest prime > n^2.
Number of ways of writing n as a sum of 7 squares.
Number of ways of writing n as a sum of 9 squares.
Number of ways of writing n as a sum of 11 squares.
Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).
Legendre symbol (n,11).
Legendre symbol (n,13).
Legendre symbol (n,17).
Legendre symbol (n,19).
Legendre symbol (n,23).
Legendre symbol (n,29).
Legendre symbol (n,31).
Legendre symbol (n,37).
Legendre symbol (n,41).
Legendre symbol (n,43).
Legendre symbol (n,47).
Legendre symbol (n,53).
Legendre symbol (n,59).
Legendre symbol (n,61).
Legendre symbol (n,67).
Legendre symbol (n,71).
Legendre symbol (n,73).
Legendre symbol (n,79).
Legendre symbol (n,83).
Legendre symbol (n,89).
Legendre symbol (n,97).
Legendre symbol (n,101).
Legendre symbol (n,103).
Legendre symbol (n,107).
Legendre symbol (n,109).
Legendre symbol (n,113).
Legendre symbol (n,127).
Legendre symbol (n,131).
Legendre symbol (n,137).
Legendre symbol (n,139).
Legendre symbol (n,149).
Legendre symbol (n,151).
Legendre symbol (n,157).
Legendre symbol (n,163).
Legendre symbol (n,167).
Legendre symbol (n,173).
Legendre symbol (n,179).
Legendre symbol (n,181).
Legendre symbol (n,191).
Legendre symbol (n,193).
Legendre symbol (n,197).
Legendre symbol (n,199).
Legendre symbol (n,211).
Legendre symbol (n,223).
Legendre symbol (n,227).
Legendre symbol (n,229).
Legendre symbol (n,233).
Legendre symbol (n,239).
Legendre symbol (n,241).
Legendre symbol (n,251).
Characteristic function of factorial numbers; also decimal expansion of Liouville’s number or Liouville’s constant.
Indices of prime Mersenne numbers (A001348).
The nonprime numbers: 1 together with the composite numbers, A002808.
Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}.
Number of partitions of n in which the greatest part is 4.
Primes congruent to {0, 2, 3} mod 5.
Squarefree values of n for which the quadratic field Q[ sqrt(n) ] is norm-Euclidean.
Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.
Primes congruent to 3 modulo 4 generated recursively: a(n) = Min_{p, prime; p mod 4 = 3; p|4Q-1}, where Q is the product of all previous terms in the sequence. The initial term is 3.
Primes of the form 6k+5 generated recursively: a(1)=5; a(n) = min{p, prime; p mod 6 = 5; p | 6Q-1}, where Q is the product of all previous terms in the sequence.
a(1)=5, a(n) is the smallest prime dividing 4*Q^2 + 1 where Q is the product of all previous terms in the sequence.
Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.
A version of the Chebyshev function theta(n): a(n) = round(Sum_{primes p <= n } log(p)).
Numbers k that divide the numerator of B(2k) (the Bernoulli numbers).
Final digit of n-th Mersenne prime A000668(n).
Mersenne primes p such that M(p) = 2^p - 1 is also a (Mersenne) prime.
Mersenne primes p such that the Mersenne number M(p) = 2^p - 1 is composite.
a(n) = prime(n^4).
G.f.: A(x) = Sum_{n>=0} (2*n+1) * 8^n * x^(n*(n+1)/2).
Numbers n such that the harmonic number numerator A001008(n) is a semiprime.
Composite numbers k such that 2^(k-4) == 1 (mod k).
Odd integers n such that 2^n == 4 (mod n).
Let A(n) = floor((3/2)^n), B(n)=3^n-2^n*A(n); then a(n)=2^n-A(n)-B(n)-2.
Composites of the form 2*n^n + 1 = A216147(n).
a(n) = n*(4*n^2 - 3*n + 5)/6.
Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.
A variant of the Euclid-Mullin sequence A000945: a(1) = 2, a(n+1) is smallest prime factor congruent to 3 (mod 4) of Product_{k=1..n} a(k) + 1.
Irregular array read by rows: n-th row contains (in ascending order) the nondivisors 1 <= k < n such that all the prime divisors p of k also divide n.
Irregular array read by rows: n-th row contains (in ascending order) the numbers 1 <= k < n such that at least one prime divisor p of k also divides n and at least one prime divisor q of k is coprime to n.
A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.
a(n) = smallest positive m such that n divides the oblong number m*(m+1).