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Old 2009-02-07, 09:48   #1
devarajkandadai
 
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May 2004

22×7×11 Posts
Default 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)
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Old 2009-02-07, 13:08   #2
devarajkandadai
 
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May 2004

4648 Posts
Default Return to failure functions

Quote:
Originally Posted by devarajkandadai View Post
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

Last fiddled with by devarajkandadai on 2009-02-07 at 13:11
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Old 2009-02-08, 04:03   #3
devarajkandadai
 
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May 2004

22·7·11 Posts
Default 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

www.crorepatibaniye.com\failurefunctions

A.K.Devaraj (To be continued)
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Old 2009-02-08, 14:54   #4
R.D. Silverman
 
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Nov 2003

22·5·7·53 Posts
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Your post should be retitled "The Failure Returns"
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