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## example: failure functions -polynomials

Let p(x) = x^2 +1 Then the failure function, when x =1, is 1 +2*k where k belongs to Z. Thus x =3,5,7,….when substituted in f(x) will yield failures i.e. composite numbers. Similarly when x=2, the failure function is 2 + 5*k.Thus 7,12, 17… , when substituted in f(x) yield failures i.e.composites congruent to 0 (mod(5)). ( Recall that our original definition of a failure is a compsite number.

## failure functions pertaining to exponential functionsL

Let f(x) be an exponential function

^{}. Let our definition of a failure continue to be a composite number. Then x = x_0 + k*Eulerphi(f(x_0)) is a failure function. Here k belongs to N. In other words values of x generated by the above failure function, when substituted in f(x), will yield only failures ( composites ).## examples of failure functions pertaining to exponential fs

Let the exp. function be 2^x+7 =f(x). When n = 2, f(x)=11.Then x=2+10*k is a failure function. In other words values of x generated by this failure function, when substitued in f(x) will yield only failures (composites) exactly divisible by 11.

## failure functions

The whole object of the messages pertaining to this subject is to give a brief proof of the infinitude of primes with the form x^2 + 1, basing the proof on a) failure functions and b) the concept of proof by inevitablity of perpetual iteration. This will be furnished shortly.