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PM outline, No\"osphere
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To continue: b) Let the mother function be 2^n + c, n and c belong to N (c is fixed). Our definition of a failure continues to be a composite number. Then n = n_0 + k*Eulerphi(f(n_0)) is a failure function. ( n_0 is a specific value of n). Here k belongs to N (To be continued).

c) Let our definition of a failure be a non-Carmichael number.Let the mother function be 2^n + 49. Note f(n) is a Carmichael number when n = 9. Then n = 5 + 6k is a failure function (k belongs to N). i.e. when we substitute values of n generated by the above failure function in the mother function we get only failures.

Let f(x) be a function of x. Then x = g(x_0) is a failure function if f(g(x_0)) is a failure in accordance with our definition of a ”failure ”. ( x_0 is a specific value of x ).

Failure functions can be applied in the following areas: a) Solving Diaphontine equations ( for copy of paper send request to b) Indirect primality testing c) In proving conjectures (see sketch proof )

Let our definition of a failure be a composite number. Let the mother function be the quadratic x^2 + 1 ( x belongs to Z ). When x =4, f(x) =17. x = 4 + 17*k is a failure function. This is linear. The non -linear failure function x = 38 + 17^(k+2) generates values of x, which when substituted in f(x) we get multiples of 289. The relevant quotients when divided by 17 yield the remainder 3, a member of Z_17. Here k belongs to W.

This refers to ” Non-linear failure functionsPlanetmathPlanetmath and AutomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The second-last line should read: When the relevant quotient is divided by 17 we get a remainder = 5, a member of Z_17.

Let our definition of a failure be a non-primitive polynomialPlanetmathPlanetmathPlanetmath in x (x belongs to Z ). Let the parent function be the primitive polynomial x^2 + x + 1. Then x generated by any of the failure functions 1 + 3k, 2 + 7k etc when substituted in the parent function yield failures i.e. non primitive polynomials.

Let our definition of a failure be a composite number which is also a multiple of 11. Let the parent function be 2^n + 7 (n belongs to N ). Then n = 2 + Eulerphi(11) is a failure function. Also n = 2^(1 + Eulerphi(Eulerphi(11)) is also a failure function.

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