## New Articles

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*Res*]**Example of stochastic matrix of mapping**by rspuzioApr 23[

*Res*]**6. Discussion**by rspuzioApr 20[

*Res*]**5. Entanglement**by rspuzioApr 20[

*Res*]**4. Measurement**by rspuzioApr 20[

*Res*]**3. Distributed dynamical systems**by rspuzioApr 20[

*Res*]**2. Stochastic maps**by rspuzioApr 20[

*Res*]**1. Introduction**by rspuzioApr 19[

*Ref*]**Oseledets multiplicative ergodic theorem**by FilipeMar 18[

*Ref*]**Furstenberg-Kesten theorem**by FilipeMar 18[

*Ref*]**multiplicative cocycle**by FilipeMar 18[

*Ref*]**remainder term series**by pahioMar 17[

*Ref*]**Structural stability theorem**by FilipeMar 15[

*Ref*]**Kac's theorem**by FilipeMar 15[

*Ref*]**Von Neumann's ergodic theorem**by FilipeMar 14## Latest Messages

5:01 am

Apr 18

Apr 14

Apr 13

Apr 12

Apr 12

Apr 6

Apr 5

Apr 3

Mar 27

Mar 20

Mar 17

Mar 12

Mar 7

Although it has been proved ( by Carl Pomerance and others) that
there are infinitely many Carmichael numbers, there can only
be finitely many Carmichael numbers a) with a given number of
primefactors and b) with a given primenumber as one of its
factors. This is evident when we apply Pomerance index (see
A 162290 on OEIS ). How to apply the Pomerance index in
arriving at upper bound for the number of Carmichael numbers
as qualified above? Steps will be furnished in subsequent messages.

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 ).

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>

## Latest Messages

5:01 am

Apr 18

Apr 14

Apr 13

Apr 12

Apr 12

Apr 6

Apr 5

Apr 3

Mar 27

Mar 20

Mar 17

Mar 12

Mar 7

Although it has been proved ( by Carl Pomerance and others) that
there are infinitely many Carmichael numbers, there can only
be finitely many Carmichael numbers a) with a given number of
primefactors and b) with a given primenumber as one of its
factors. This is evident when we apply Pomerance index (see
A 162290 on OEIS ). How to apply the Pomerance index in
arriving at upper bound for the number of Carmichael numbers
as qualified above? Steps will be furnished in subsequent messages.

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 ).

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>