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additive number theory


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1: 27.1 Special Notation
§27.1 Special Notation
d , k , m , n

positive integers (unless otherwise indicated).

2: 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
3: 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
4: 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.

  • 5: Bibliography V
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.
  • D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ (1988) Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.
  • N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
  • M. N. Vrahatis, O. Ragos, T. Skiniotis, F. A. Zafiropoulos, and T. N. Grapsa (1997b) The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.