# 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

• (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.

• (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.

• (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 FailureFunctionTests 2013-03-22 19:33:30 2013-03-22 19:33:30 bci1 (20947) bci1 (20947) 14 bci1 (20947) msc 00-02 msc 00-01