Return to failure functions

[devarajkandadai]

Old timers may perhaps recall my posts pertaining to failure functions. I now propose to present in a sysematic manner the definitions pertaining to
the above relevant to the following areas of number theory: a) Polynomial functions b) exponential functions and Diophantine equations.


Polynomial functions

Let phi(x) be a function of x ( x belongs to Z). Let the definition of a failure be a composite number. Then x = psi(x_0) = x_0 + k.phi(x_0)
generate values of x which enable phi(x) to generate only failures (composite numbers). Here x_0 is a specific value of x and k belongs to N.

Let me give a simple numerical illustration. Let phi(x) = x^2 + x + 15.
When x =1 phi(x) = 17. x = psi(1) = 1 + k.17 generates values of x which when substituted in phi(x) yield only composite numbers (each a multiple of 17).

Note: when phi(x) is composite each factor contributes a failure function.
A.K.Devaraj (To be continued)

[devarajkandadai]

Originally Posted by devarajkandadai

Old timers may perhaps recall my posts pertaining to failure functions. I now propose to present in a sysematic manner the definitions pertaining to
the above relevant to the following areas of number theory: a) Polynomial functions b) exponential functions and Diophantine equations.


Polynomial functions

Let phi(x) be a function of x ( x belongs to Z). Let the definition of a failure be a composite number. Then x = psi(x_0) = x_0 + k.phi(x_0)
generate values of x which enable phi(x) to generate only failures (composite numbers). Here x_0 is a specific value of x and k belongs to N.

Let me give a simple numerical illustration. Let phi(x) = x^2 + x + 15.
When x =1 phi(x) = 17. x = psi(1) = 1 + k.17 generates values of x which when substituted in phi(x) yield only composite numbers (each a multiple of 17).

Note: when phi(x) is composite each factor contributes a failure function.
A.K.Devaraj (To be continued)

To continue:



Exponential functions:

Let phi(x)= a^x + c where a,x and c belong to N, a and c being fixed.



Let the definition of a failure again be a composite number. Then x = psi(x_0) = x_0 + k.Eulerphi(phi(x_0)) is a failure function since phi(psi(x_0)) generates only failures ( composite numbers).

Note: phi(psi(x_0)) are multiples of phi(x_0).

Numerical illustration: Let phi(x) = 2^n + 7.

When x =1, phi(1) = 9 and x = psi(1) = 1 + k.Eulerphi(9) is a failure function generating values of x such that phi(x) are failures (composites) being multiples of 9. When x=2, phi(2) = 11and the relevant failure function is x = psi(2) = 2 + k.10.

i.e. phi(x) for values of x = 2, 12, 22....generates only failures (all multiples of 11).

Pl note a) whenever phi(x) is composite each factor contributes a failure function and b) this is a generalisation of Fermat's theorem.
(To be continued)

A.K.Devaraj

[devarajkandadai]

c. Diohantine equations

Perhaps the best intro to the role of failure functions in solving Diophantine equations wd be my paper " A Theorem a la Ramanujan" on

www.crorepatibaniye.com\failurefunctions

A.K.Devaraj (To be continued)

[R.D. Silverman]

Your post should be retitled "The Failure Returns"