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PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community. On February 13th 2013, was updated to use the new software system Planetary. Some release notes are here. Please report bugs in the Planetary Bugs Forum or on Github.

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Latest Messages  

[P] failure functions - remarks by akdevaraj Apr 18
Failure functions predict the values of the variable, which when substituted in the mother function, yield failures, in accordance with our definition of a failure. Normally we cannot say anything about the the other values of the variable - they may yield failures or successes. However in the case of the quadratic x^2 + 1 we can definitely say that the values of of x not covered by the failure functions yield only successes - (see sketch proof ).

[P] failure functions - another example by akdevaraj Apr 14
Let our definition of a failure be a non - Devarajnumber which is not a Carmichael number ( see A104017 on OEIS ).Let the mother function be 2^n + 3113. Then n = 16 + 42*k is a failure function ).

[P] Failure functions - role of by akdevaraj Apr 13
failure functions play an important role in proving a conjecture indirectly. See sketch proof.

[P] failure functions - another example by akdevaraj Apr 12
Here k belongs to N.When n= 10, f(n) = 1105, a Carmichael number; however when n is generated by the failure function 18 + 20*k we get values of f(n) which are not square free and hence incapable of being Carmichael numbers i.e. failures.

[P] failure functions - another example by akdevaraj Apr 12
Let our definition Let our definition of a failure be a non-Carmichael number. Let the mother function be 2^n + 81. Then n = 18 + 20*k is a failure function.

[P] Fermat's theorem by akdevaraj Apr 6
A Bit of History: Fermat's little theorem --1640 Euler's generalisation of Fermat's theorem - circa 1740 Euler's generalisation of Fermat's theorem - a further generalisation (DEvaraj) - 2004 Ultimate generalisation of Fermat's theorem ( Pahio and Devaraj ) - 2012

[P] A puzzle by akdevaraj Apr 5
About a year ago I had given a sketch proof of the infinitude of primes having the form x^2 +1 (using failure functions ). Today when I searched for" sketch proof " the search did not reveal this topic Wonder whether this has been deleted, Could anyone pl enlighten me? A.K. Devaraj

[P] cyrillics already visible by pahio Apr 3
Who has helped? Thanks! All cyrillics in are today visible (except the \cyry which looks mysterious). Jussi

[P] Cyrillics not visible by pahio Mar 27
Hi unlord and other experts, Why the Cyrillic letters in the References of don't work? In an older entry they work well. Both have similar preambles. Regards, Jussi

[P] math processing error by lichen Mar 20
Why the web papge appeared meesage "math processing error "

[p] Re: How to attach a file to an article by jac Mar 17
Hi Robert: There's no immediately straightforward way to do that, sorry! The "PlanetMath way" would be to create a second article (Code to accompany XYZ article), and put the title of the original article in the Parent field. Use \verb|\begin{verbatim}| and \verb|\end{verbatim}| tags at the beginning and end of the code article to preserve whitespace formatting. Joe

[P] Is a^0 an empty product or not? by hoeij Mar 12
a^0 is an empty product for every a (not just non-zero a). After all, an empty set that contains no a=3's is the same as an empty set that contains no a=0's.

How to attach a file to an article by robert_dodier Mar 7
Hi, I've written an article and I'd like to attach a file (a program) to the article. I couldn't figure out how to do that -- any ideas? best, Robert Dodier>

[p] They always exist provided by Filipe Mar 5
They always exist provided you consider the extended real line, i.e, adding the two infites to the line. They exist because, for instance, lim inf is an increasing sequence, that is bounded from above (by infinity). So it converges (eventually it may converge to infinity). The same argument is valid for lim sup, which is decreasing