failure function tests
1 FAILURE FUNCTION
An abstract definition is:
Let $\varphi (x)$ be a function^{} of $x$. Then, $x=\psi ({x}_{0})$ is a failure function if the values of $x$ generated by $\psi ({x}_{0})$, when substituted in $\varphi (x)$, generate only failures in accordance with our definition of a failure. Here ${x}_{0}$ is a specific value of $x$.
1.1 Examples

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(i) Let the mother function be a polynomial^{} in x (coeffficients belong to $\mathcal{Z}$ ), say $\varphi (x)$. Let our definition of a failure be a composite number^{}. Then, $x=\psi ({x}_{0})={x}_{0}+k(\varphi ({x}_{0}))$ is a failure function because the values of x generated by $\varphi ({x}_{0})$, when substituted in $\varphi (x)$ , generate only failures.

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(ii) Let the mother function be an exponential function^{}, say $\varphi (x)={a}^{x}+c$. Then $x=\psi ({x}_{0})={x}_{0}+k.Eulerphi(\varphi ({x}_{0}))$ is a failure function since the values of x generated by $\psi ({x}_{0})$, when substituted in the mother function, generate only failures.

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(iii) Let our definition of a failure be a nonCarmichael number. Let the mother function $\varphi (x)$ be ${2}^{n}+49$. Then, $n=5+6k$ is its failure function $\psi (x)$.
1.2 Note
Here too our definition of a failure is a composite number and k belongs to N.
Title  failure function tests 

Canonical name  FailureFunctionTests 
Date of creation  20130322 19:33:30 
Last modified on  20130322 19:33:30 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  14 
Author  bci1 (20947) 
Classification  msc 0002 
Classification  msc 0001 