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## Latest Messages

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 ).

Apr 13
failure functions play an important role in proving a conjecture indirectly. See sketch proof.

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.

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.

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

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

Apr 3
Who has helped? Thanks! All cyrillics in http://planetmath.org/remaindertermseries are today visible (except the \cyry which looks mysterious). Jussi

Mar 27
Hi unlord and other experts, Why the Cyrillic letters in the References of http://planetmath.org/remaindertermseries don't work? In an older entryhttp://planetmath.org/divisorasfactorofprincipaldivisor they work well. Both have similar preambles. Regards, Jussi

Mar 20
Why the web papge appeared meesage "math processing error "

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

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.

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>

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

[p] Solution by Filipe Mar 5
The expectation E(X)-E(min(X,M)) can be written $$\sum_x (x-min(x,M))p(x)$$ Instead of summing over all the values of the random variable X, you can sum just over the values of the random variable for which x-min(x,M) is strictly positive. Then you must have that the probability of the random variable X taking such a x is zero. So the probability $P(X > M)$ must be zero.