primes
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1: 27.18 Methods of Computation: Primes
§27.18 Methods of Computation: Primes
►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer is given in Crandall and Pomerance (2005, §3.7). … ►The Sieve of Eratosthenes (Crandall and Pomerance (2005, §3.2)) generates a list of all primes below a given bound. … ►For small values of , primality is proven by showing that is not divisible by any prime not exceeding . … ►The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …2: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
… ►Prime Number Theorem
… ►The number of such primes not exceeding is … ►A Mersenne prime is a prime of the form . The largest known prime (2018) is the Mersenne prime . …3: 29.10 Lamé Functions with Imaginary Periods
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►The first and the fourth functions have period ; the second and the third have period .
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29.10.2
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29.10.3
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4: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. … ►If are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer has prime factorization , then the Jacobi symbol is defined by , with . …Both (27.9.1) and (27.9.2) are valid with replaced by ; the reciprocity law (27.9.3) holds if are replaced by any two relatively prime odd integers .5: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
( is defined to be 0.)
…Tables of primes (§27.21) reveal great irregularity in their distribution.
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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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§27.2(ii) Tables
…6: 27.1 Special Notation
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positive integers (unless otherwise indicated). | |
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greatest common divisor of . If , and are called relatively prime, or coprime. | |
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sum taken over , and relatively prime to . | |
prime numbers (or primes): integers () with only two positive integer divisors, and the number itself. | |
, | sum, product extended over all primes. |
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7: 27.3 Multiplicative Properties
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►Except for , , , and , the functions in §27.2 are multiplicative, which means and
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►If is multiplicative, then the values for are determined by the values at the prime powers.
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27.3.2
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27.3.5
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27.3.10
8: 10.58 Zeros
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►For the th positive zeros of , , , and are denoted by , , , and , respectively, except that for we count as the first zero of .
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►For some properties of and , including asymptotic expansions, see Olver (1960, pp. xix–xxi).
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9: 16.13 Appell Functions
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16.13.1
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16.13.2
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16.13.3
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16.13.4
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►Here and elsewhere it is assumed that neither of the bottom parameters and is a nonpositive integer.
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