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PlanetMath
PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigor, openness, pedagogy, realtime content, interlinked content, and community. Intended to be comprehensive, the project is located at the Digital Library Research Lab at Virginia Tech.
PlanetMath was started when the popular free online mathematics encyclopedia MathWorld was taken offline by a court injunction as a result of the CRC Press lawsuit against the Wolfram Research company and its employee (and MathWorld’s author) Eric Weisstein.
PlanetMath uses the same copyleft as Wikipedia: the GNU Free Documentation License. An author who starts a new article becomes the owner of that article; he or she may then choose to grant editing rights to other individuals or groups. All textual content and mathematical formulas are written in LaTeX, a typesetting system that requires some learning but is popular among mathematicians because of its support of the technical needs of mathematical typesetting and its highquality output. The user can explicitly create links to other articles, and the system also automatically turns certain words into links to the defining articles. For more details on the automatic linking, see the collaboration on PlanetMath automatic reference linking. For more details on controlling the linking of an article, see the collaboration on controlling linking The topic area of every article is classified by the Mathematics Subject Classification of the American Mathematical Society. Users may attach addenda, errata, and discussions to articles.
The most common method of public communication within PlanetMath is posts. Users can add posts in the forums as well as attach posts to articles, corrections, collaborations, requests for new articles, and other posts. A system for private messaging among users is also in place.
Users who are new to PlanetMath are highly encouraged to read the following collaborations:
The software running PlanetMath is written in Perl and runs on Linux and the Apache Web server. It is known as $No\"{o}sphere$ and has been released under the free BSD License.
Most of the very most basic topics are covered, though PlanetMath is striving to improve coverage of elementary and intermediate topics. Due to the increasing popularity of the package PSTricks, more members of PlanetMath are able to incorporate graphics into their articles. This has enabled PlanetMath to cover many elementary and intermediate topics in geometry that were once lacking. There are several methods of creating graphics on PlanetMath. For more details on creating graphics on PlanetMath, see the collaboration on graphics and PlanetMath.
PlanetMath also has entries on highly advanced and specialized topics. PlanetMath has entries on the integers 42 and 666. The following toplevel Mathematics Subject Classification categories have only one or two topic entries at PlanetMath:

74XX, mechanics of deformable solids;

76XX, fluid mechanics;

85XX, astronomy and astrophysics.
The Wikipedia:WikiProject Mathematics/PlanetMath Exchange project assists in content exchange between PlanetMath and Wikipedia.
This entry was adapted from the Wikipedia article PlanetMath as of February 24, 2007.
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failure functions  refresher  II
To continue: b) Let the mother function be 2^n + c, n and c belong to N (c is fixed). Our definition of a failure continues to be a composite number. Then n = n_0 + k*Eulerphi(f(n_0)) is a failure function. ( n_0 is a specific value of n). Here k belongs to N (To be continued).
Failure functions  refresher III
c) Let our definition of a failure be a nonCarmichael number.Let the mother function be 2^n + 49. Note f(n) is a Carmichael number when n = 9. Then n = 5 + 6k is a failure function (k belongs to N). i.e. when we substitute values of n generated by the above failure function in the mother function we get only failures.
failure functions  refresher  iv  abstract definition
Let f(x) be a function of x. Then x = g(x_0) is a failure function if f(g(x_0)) is a failure in accordance with our definition of a ”failure ”. ( x_0 is a specific value of x ).
failure functions  applications
Failure functions can be applied in the following areas: a) Solving Diaphontine equations ( for copy of paper send request to dkandadai@gmail.com b) Indirect primality testing c) In proving conjectures (see sketch proof )
Nonlinear failure functions and automorphism
Let our definition of a failure be a composite number. Let the mother function be the quadratic x^2 + 1 ( x belongs to Z ). When x =4, f(x) =17. x = 4 + 17*k is a failure function. This is linear. The non linear failure function x = 38 + 17^(k+2) generates values of x, which when substituted in f(x) we get multiples of 289. The relevant quotients when divided by 17 yield the remainder 3, a member of Z_17. Here k belongs to W.
A correction
This refers to ” Nonlinear failure functions^{} and Automorphism^{}. The secondlast line should read: When the relevant quotient is divided by 17 we get a remainder = 5, a member of Z_17.
failure functions  another example
Let our definition of a failure be a nonprimitive polynomial^{} in x (x belongs to Z ). Let the parent function be the primitive polynomial x^2 + x + 1. Then x generated by any of the failure functions 1 + 3k, 2 + 7k etc when substituted in the parent function yield failures i.e. non primitive polynomials.
failure functions  another exampleL
Let our definition of a failure be a composite number which is also a multiple of 11. Let the parent function be 2^n + 7 (n belongs to N ). Then n = 2 + Eulerphi(11) is a failure function. Also n = 2^(1 + Eulerphi(Eulerphi(11)) is also a failure function.