20070809, 05:56  #1 
May 2004
2^{2}·79 Posts 
Failure Functions, Episode IX, Revenge of the Sith
I paste below a brief note on failure functions and hope researchers find it useful.
FAILURE FUNCTIONS ( A brief note). Abstract definition of failure functions: Let f(x) be a function of x.Then x=psi(x_o ) is a failure function if f(psi(x_o)) is a failure in accordance with our definition of a failure. Examples: a)Polynomials: Let f(x) be a polynomial in x (x belongs to C).Then x= x_o + f(x_o) is a failure function.This can be easily proved by Taylor's Theorem. b)Exponential functions. Let f(x) be an exponential function.Then x= x_o + k*Eulerphi(f(x_o) is a failure(.Here a "failure" is defined as a composite number.k belongs to Z in the case of polynomials and it belongs to N in the case of exponential functions (example:2^n+7).x_o is a specific value of x. Note: i) The definition of a failure is arbirary.However, once it has been defined, we must be able to formulate the relevant failure function. ii) In the examples citedvariable pertaining to polynomials belongs to C.In the case of exponential functions the variable belongs to N. iii) Researchers in various fields such as Operations Research(Linear Programming),Computer science (search techniques),group theory etc. can perhaps apply the concept. iv) A demonstration of application in a proof can be seen in "A Theorem a la Ramanujan" (www.crorepatibaniye.com/failurefunctions). 
20070809, 13:20  #2 
"William"
May 2003
New Haven
3·787 Posts 
This "brief note" in isolation is utterly meaningless. It defines a failure function using THREE undefined terms:
1. "our definition of failure" 2. x_0 3. psi() 
20070809, 16:57  #3 
∂^{2}ω=0
Sep 2002
República de California
10110101101001_{2} Posts 
Mr. Devaraj already has multiple threads about his famous "failure functions" in Misc. Maths:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 so moving this one there, as well. Last fiddled with by ewmayer on 20070809 at 17:09 Reason: "Full of sound and fury..." 
20070812, 03:01  #4 
Jun 2003
Pa.,U.S.A.
2^{2}·7^{2} Posts 
comment on deemed failure functions
I have a tendency to want to rub my hands with excitement over this,even though my hands are topologically chiral,and I haven't delved into the relations being considered in full.

20070812, 03:52  #5 
Jun 2005
USA, IL
193 Posts 
whoa! get the ointment!

20070812, 04:09  #6  
"Mike"
Aug 2002
1F63_{16} Posts 
It is midnight and here we are, mysteriously channeling Ernst, the coprologic commentator:
Quote:


20070813, 05:35  #7  
Bronze Medalist
Jan 2004
Mumbai,India
100000000100_{2} Posts 
An open question to Mr. Devraj!
Quote:
Well Ernst at least you have bunched the lot together which makes easy reference to the various threads he has posted. 1) Dev can you please tell me if the first paper on "failure function" is your original work as the others I presume are based on it ? 2) What practical utility is it in Number theory? 3) Is it good for finding primes or OTOH is it meant to find composites? 4) Can it show the way to easy factoring of large numbers and if so then how? 5) You have said somewhere that all that is required is a calculator. Is this true? 6) Does it require division at every step ? 7)Has it got an easy sieving method to determine primes? 8) Why do you call it "failure" ? I would be highly obliged if you answer all so that I for one, and the rest of us, can go more deeply into your theories. I'm sorry to ask these silly questions as you have personally explained the principle a long time ago and I have forgotten, though I still have your copies.with me intact in my archives. Kindly bear with me Dev as I am still a novice in Number theory! I have lost contact with you for several years now. Kindly contact me by phone. Thank you Mally 

20070821, 18:16  #8 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Failure Functions!
[QUOTE=ewmayer;112061]Mr. Devaraj already has multiple threads about his famous "failure functions" in Misc. Maths: [QUOTE]
It will be interesting to know and maybe a surprise for Mr. Devraj that his "failure functions" is not an original and was well known to Oystein Ore' and published by him as far back as 1948 in his book "Number Theory and its History". Ore did some original work and made a conjecture which has not been proved one way or the other. Perhaps Hardy and Ramanujan also knew of this function way back in the 30's. Well Ore' devotes just two pages to it and I can quote him verbatim as I have the book. As far as I can undestand it this is one and the same as 'Failure functions'. It is a way to determine composites AFAIK. Well that sounds silly to me right now but I'm afraid thats what it is as I'm writing ad lib and from memory. But the coincidence seemed to me as odd If my statement is contested I will take the trouble to reproduce Ore's two pages and his worked example. The theory is old but the name is knew! All the same I would like to commend Mr. Devaraj though I have not followed his other threads but I do know that he has extended it and developed it admirably well and produced some original results. Keep up the good work Dev. I hope I am wrong. Mally "coffee: 
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