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

• (i) Let the mother function be a polynomialin x (coeffficients belong to $𝒵$ ), 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.

• (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.

• (iii) Let our definition of a failure be a non-Carmichael 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.