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Failure functions

Type of Math Object: 
Definition
Major Section: 
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Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

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.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure function if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=xpsisubscriptx0absentx=psi(x_{0})=x_0 +k*fragmentsk+k*ϕ(x_0)isafailurefunctionsinceisafailurefunctionsinceisafailurefunctionsinceϕ(psi(x_0))willgenerateonlyfailures(composites).2):Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,fragmentswillgenerateonlyfailuresfragmentsnormal-(compositesnormal-).2normal-)normal-:Letourdefinitionofafailureagainbeacompositenumbernormal-.Lettheparentfunctionnormal-,willgenerateonlyfailures(composites).2):% Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,ϕ(x)beanexponentialfunction.Thenformulae-sequencebeanexponentialfunctionThenbeanexponentialfunction.Thenx==ψ(x_0) = x_0 + Eulerphi(ϕ(x_0))isafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites).3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbefragmentsisafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailuresfragmentsnormal-(compositesnormal-).3normal-)normal-:LetourdefinitionofafailurebeanonCarmichaelnumbernormal-.Lettheparentfunctionbeisafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites% ).\par 3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbe2^n + 49.Thenfragmentsnormal-.Then.Then n= 5 + 6*kisafailurefunction.Hereformulae-sequenceisafailurefunctionHereisafailurefunction.HerekbelongstobelongstobelongstoN.Applications:fragmentsnormal-.Applicationsnormal-:.\par Applications:a)indirectprimalitytestingandindirectprimalitytestingandindirectprimalitytestingandb)inprovingtheoremsinnumbertheory.inprovingtheoremsinnumbertheoryinprovingtheoremsinnumbertheory.\@add@PDF@RDFa@triples\end{document}

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ψxpsisubscriptx0subscriptx0kψx=psi(x_{0})=x_{0}+k*\psi is a failure function since ϕ(psi(x0))ϕpsisubscriptx0\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, ϕ(x)ϕx\phi(x) be an exponential function. Then xxx = ψ(x0)=x0+Eulerphi(ϕ(x0))ψsubscriptx0subscriptx0Eulerphiϕsubscriptx0\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 2n+49superscript2n492^{n}+49. Then n=5+6*kn56kn=5+6*k is a failure function. Here kkk belongs to NNN.

Applications: a)fragmentsanormal-)a) indirect primality testing and b)fragmentsbnormal-)b) in proving theorems in number theory.

Background: see messages.

Abstract definition: let phi(x)phixphi(x) be a function of xxx. Then x=psi(x0)xpsisubscriptx0x=psi(x_{0}) is a failure if phi(psi(x0))phipsisubscriptx0phi(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, ϕ(x)ϕx\phi(x), be a polynomial ring where the variable and coefficients belong to ZZZ. Then x=psi(x0)=x0+k*ϕ(x0)isafailurefunctionsincexpsisubscriptx0subscriptx0kϕsubscriptx0isafailurefunctionsincex=psi(x_{0})=x_{0}+k*\phi(x_{0})isafailurefunctionsinceϕ(psi(x_0))willgenerateonlyfailures(composites).2):Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,fragmentswillgenerateonlyfailuresfragmentsnormal-(compositesnormal-).2normal-)normal-:Letourdefinitionofafailureagainbeacompositenumbernormal-.Lettheparentfunctionnormal-,willgenerateonlyfailures(composites).2):% Letourdefinitionofafailureagainbeacompositenumber.Lettheparentfunction,ϕ(x)beanexponentialfunction.Thenformulae-sequencebeanexponentialfunctionThenbeanexponentialfunction.Thenx==ψ(x_0) = x_0 + Eulerphi(ϕ(x_0))isafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites).3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbefragmentsisafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailuresfragmentsnormal-(compositesnormal-).3normal-)normal-:LetourdefinitionofafailurebeanonCarmichaelnumbernormal-.Lettheparentfunctionbeisafailurefunctionsincetheparentfunctionwillnowgenerateonlyfailures(composites% ).\par 3):Letourdefinitionofafailurebeanon-Carmichaelnumber.Lettheparentfunctionbe2^n + 49.Thenfragmentsnormal-.Then.Then n= 5 + 6*kisafailurefunction.Hereformulae-sequenceisafailurefunctionHereisafailurefunction.HerekbelongstobelongstobelongstoN.Applications:fragmentsnormal-.Applicationsnormal-:.\par Applications:a)indirectprimalitytestingandindirectprimalitytestingandindirectprimalitytestingandb)inprovingtheoremsinnumbertheory.inprovingtheoremsinnumbertheoryinprovingtheoremsinnumbertheory.\@add@PDF@RDFa@triples\end{document}

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.

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) .

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) .

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