Failure functions

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failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

correction: The fifth line of abstract definition should read ” x = psi(x_0) = x_0 + k*phi(x_0) ”. Here k belongs to Z Also line eight of abstract definition should read x = psi (x_0) = x_0 + k*Eulerphi(x_0); here k belongs to N.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure function if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=$x_0 $+k*$ϕ(x_0)$isafailurefunctionsince$ϕ(psi(x_0))$willgenerateonlyfailures(composites).2):% Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,$ϕ(x)$beanexponentialfunction.Then$x$=$ψ(x_0) = x_0 + Eulerphi(ϕ(x_0))$isafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites% ).\par 3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbe$2^n + 49$.Then$ n= 5 + 6*k$isafailurefunction.Here$k$belongsto$N$.\par Applications:$a)$indirectprimalitytestingand$b)$inprovingtheoremsinnumbertheory.\@add@PDF@RDFa@triples\end{document}$

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\psi$ is a failure function since $\phi(psi(x_{0}))$ will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function, $\phi(x)$ be an exponential function. Then $x$ = $\psi(x_{0})=x_{0}+Eulerphi(\phi(x_{0}))$ is a failure function since the parent function will now generate only failures ( composites).

3): Let our definition of a failure be a non-Carmichael number. Let the parent function be $2^{n}+49$. Then $n=5+6*k$ is a failure function. Here $k$ belongs to $N$.

Applications: $a)$ indirect primality testing and $b)$ in proving theorems in number theory.

failure functions

Background: see messages.

Abstract definition: let $phi(x)$ be a function of $x$. Then $x=psi(x_{0})$ is a failure if $phi(psi(x_{0}))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x=psi(x_{0})=x_{0}+k*\phi(x_{0})isafailurefunctionsince$ϕ(psi(x_0))$willgenerateonlyfailures(composites).2):% Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,$ϕ(x)$beanexponentialfunction.Then$x$=$ψ(x_0) = x_0 + Eulerphi(ϕ(x_0))$isafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites% ).\par 3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbe$2^n + 49$.Then$ n= 5 + 6*k$isafailurefunction.Here$k$belongsto$N$.\par Applications:$a)$indirectprimalitytestingand$b)$inprovingtheoremsinnumbertheory.\@add@PDF@RDFa@triples\end{document}$

failure functions - another example

Let our definition of a failure be a non-square free number. Let the parent function be a polynomial, say f(x) =x^2+x+1. This function generates a falure for certain values of x. Example x = 18. 18^2 + 18 + 1 = 343 = 7^3. Then x = 18 + k*343 is a failure function ( here k belongs to Z) .In other words any value of x generated by this failure function, when substtuted in the parent function we get a failure in accordance with above definition of a failure.

failure functions - another example

Let our definition of a failure be a non-square free number. Let the parent function be a polynomial, say f(x) =x^2+x+1. This function generates a falure for certain values of x. Example x = 18. 18^2 + 18 + 1 = 343 = 7^3. Then x = 18 + k*343 is a failure function ( here k belongs to Z) .

failure functions - another example

Let our definition of a failure be a non-square free number. Let the parent function be a polynomial, say f(x) =x^2+x+1. This function generates a falure for certain values of x. Example x = 18. 18^2 + 18 + 1 = 343 = 7^3. Then x = 18 + k*343 is a failure function ( here k belongs to Z) .