failure function tests

1 FAILURE FUNCTION

An abstract definition is:

Let $\phi(x)$ 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 $\phi(x)$ , 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 $\phi(x)$ . Let our definition of a failure be a composite number. Then, $x=\psi(x_{0})=x_{0}+k(\phi(x_{0}))$ is a failure function because the values of x generated by $\phi(x_{0})$ , when substituted in $\phi(x)$ , generate only failures.
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(ii) Let the mother function be an exponential function, say $\phi(x)=a^{x}+c$ . Then $x=\psi(x_{0})=x_{0}+k.Eulerphi(\phi(x_{0}))$ 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 non-Carmichael number. Let the mother function $\phi(x)$ 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	2013-03-22 19:33:30
Last modified on	2013-03-22 19:33:30
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	14
Author	bci1 (20947)
Classification	msc 00-02
Classification	msc 00-01