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21: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
►Except for , , , and , the functions in §27.2 are multiplicative, which means and … ►
27.3.2
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27.3.6
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27.3.10
22: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
… ►Prime Number Theorem
… ►The largest known prime (2018) is the Mersenne prime . … ►There are infinitely many Carmichael numbers.23: 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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24: 24.6 Explicit Formulas
25: 27.13 Functions
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§27.13(i) Introduction
… ►§27.13(ii) Goldbach Conjecture
… ►§27.13(iii) Waring’s Problem
… ►For a given integer the function is defined as the number of solutions of the equation … ►Hence because both divisors, and , are congruent to . …26: 24.20 Tables
§24.20 Tables
►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D. ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. In Wagstaff (2002) these results are extended to and , respectively, with further complete and partial factorizations listed up to and , respectively. …27: 24.9 Inequalities
§24.9 Inequalities
►Except where otherwise noted, the inequalities in this section hold for . … ►
24.9.2
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►(24.9.3)–(24.9.5) hold for .
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►(24.9.6)–(24.9.7) hold for .
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28: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. …If does not divide , then has the value when the quadratic congruence has a solution, and the value when this congruence has no solution. … ►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 . …29: 4.19 Maclaurin Series and Laurent Series
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►In (4.19.3)–(4.19.9), are the Bernoulli numbers and are the Euler numbers (§§24.2(i)–24.2(ii)).
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4.19.3
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4.19.4
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4.19.8
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4.19.9
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30: 27.19 Methods of Computation: Factorization
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►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored.
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►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard , and a 67-digit prime for ecm.
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►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General
Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs).
…The snfs can be applied only to numbers that are very close to a power of a very small base.
The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs.
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