failure function tests


1 FAILURE FUNCTION

An abstract definition is:

Let ϕ(x) be a functionMathworldPlanetmath of x. Then, x=ψ(x0) is a failure function if the values of x generated by ψ(x0), when substituted in ϕ(x), generate only failures in accordance with our definition of a failure. Here x0 is a specific value of x.

1.1 Examples

  • (i) Let the mother function be a polynomialPlanetmathPlanetmath in x (coeffficients belong to 𝒵 ), say ϕ(x). Let our definition of a failure be a composite numberMathworldPlanetmath. Then, x=ψ(x0)=x0+k(ϕ(x0)) is a failure function because the values of x generated by ϕ(x0), when substituted in ϕ(x) , generate only failures.

  • (ii) Let the mother function be an exponential functionDlmfDlmfMathworld, say ϕ(x)=ax+c. Then x=ψ(x0)=x0+k.Eulerphi(ϕ(x0)) is a failure function since the values of x generated by ψ(x0), 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 ϕ(x) be 2n+49. Then, n=5+6k is its failure function ψ(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