$d, k, m, n$	positive integers (unless otherwise indicated).
…
$p,p_{1},p_{2},\dots$	prime numbers (or primes): integers ( $>1$ ) with only two positive integer divisors, $1$ and the number itself.
…
$x, y$	real numbers.
…

5: 27.13 Functions

… ►

§27.13(i) Introduction

… ►The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

… ►

§27.13(iii) Waring’s Problem

… ►

§27.13(iv) Representation by Squares

…

6: 27.22 Software

… ►In this section we provide links to the known sources of software for factorization and primality testing, as well as additional Web-based resources for information on these topics. … ►

•

Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below $10^{16}$ . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$ , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

… ►

•

Number Theory Web. References and links to software for factorization and primality testing.

…

7: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

8: 6.16 Mathematical Applications

… ►See Carslaw (1930) for additional graphs and information. … ►

§6.16(ii) Number-Theoretic Significance of $\operatorname{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then …where

\pi(x)

is the number of primes less than or equal to

x

. … ►

See accompanying text — Figure 6.16.2: The logarithmic integral $\operatorname{li}\left(x\right)$ , together with vertical bars indicating the value of $\pi(x)$ for $x=10,20,\dots,1000$ . Magnify

9: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that … ►

§27.14(iv) Relation to Modular Functions

… ►

§27.14(v) Divisibility Properties

… ►

§27.14(vi) Ramanujan’s Tau Function

… ►

10: 25 Zeta and Related Functions

…

additive%20number%20theory

1—10 of 385 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

2: 8 Incomplete Gamma and Related Functions

3: 23 Weierstrass Elliptic and Modular Functions

4: 27.1 Special Notation

§27.1 Special Notation