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21: 27.1 Special Notation
§27.1 Special Notation
… ►positive integers (unless otherwise indicated). | |
… | |
prime numbers (or primes): integers () with only two positive integer divisors, and the number itself. | |
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real numbers. | |
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22: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
… ►Prime Number Theorem
… ►The number of such primes not exceeding is … ►There are infinitely many Carmichael numbers.23: 27.2 Functions
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§27.2(i) Definitions
… ►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing . … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►§27.2(ii) Tables
…24: 24.5 Recurrence Relations
§24.5 Recurrence Relations
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24.5.3
,
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24.5.5
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§24.5(ii) Other Identities
… ►§24.5(iii) Inversion Formulas
…25: 24.6 Explicit Formulas
26: 27.13 Functions
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§27.13(i) Introduction
►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►§27.13(ii) Goldbach Conjecture
… ►§27.13(iii) Waring’s Problem
…27: Bibliography
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Congruences of -adic integer order Bernoulli numbers.
J. Number Theory 59 (2), pp. 374–388.
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Sharp bounds for the Bernoulli numbers.
Arch. Math. (Basel) 74 (3), pp. 207–211.
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Number Theory.
In The New Encyclopaedia Britannica,
Vol. 25, pp. 14–37.
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A Centennial History of the Prime Number Theorem.
In Number Theory,
Trends Math., pp. 1–14.
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A primer on Bernoulli numbers and polynomials.
Math. Mag. 81 (3), pp. 178–190.
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28: 26.13 Permutations: Cycle Notation
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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►A permutation is even or odd according to the parity of the number of transpositions.
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29: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. … ►
27.9.2
►If are distinct odd primes, then the quadratic reciprocity law states that
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►If an odd integer has prime factorization , then the Jacobi symbol
is defined by , with .
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30: 24.21 Software
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