Return to failure functions
 

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)

Return to failure functions
 

[QUOTE=devarajkandadai;161965]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)[/QUOTE]


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

Return to failure functions
 

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

[url]www.crorepatibaniye.com\failurefunctions[/url]

A.K.Devaraj (To be continued)

Your post should be retitled "The Failure Returns"