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PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community.

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## Latest Messages

May 29
We don't have a specific policy about this, more of a long-standing tradition. It is better to have one article on a topic, and add co-authors, rather than several. At some point we may change this approach. Akdevaraj, can you point me to the articles in question? -Joe PS. I'll also respond to your email soon, sorry about the delay.

May 29
A friend suggested that I complain about the fact that some member had misappropriated my intellectual property: "Failure functions". My point: mathematics is such a vast subject - there is no need for anyone to misappropriate some other member's contribution. Everyday one can discover some new aspect of number theory, group theory algebraic geometry etc.

May 27
Background: see messages.\newline Abstract definition: let $phi(x)$ be a function of $x$. Then $x = psi(x_0)$ is a failure function if $phi(psi(x_0))$ is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function, $\phi(x)$, be a polynomial ring where the variable and coefficients belong to $Z$. Then $x = psi(x_0) =$x_0 $+ k*$\phi(x_0)$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 27 Background: see messages.\newline Abstract definition: let$phi(x)$be a function of$x$. Then$x = psi(x_0)$is a failure if$phi(psi(x_0))$is a failure in accordance with our definition of a failure. Examples: 1) Let our definition of a failure be a composite number. Let the parent function,$\phi(x)$, be a polynomial ring where the variable and coefficients belong to$Z$. Then$x = psi(x_0) =x_0 + k*\psi$is a failure function since$\phi(psi(x_0))$will generate only failures (composites). 2): Let our definition of a failure again be a composite number. Let the parent function,$\phi(x)$be an exponential function. Then$x$=$\psi(x_0) = x_0 + Eulerphi(\phi(x_0))$is a failure function since the parent function will now generate only failures ( composites). 3): Let our definition of a failure be a non-Carmichael number. Let the parent function be$2^n + 49$. Then$ n= 5 + 6*k$is a failure function. Here$k$belongs to$N$. Applications:$a)$indirect primality testing and$b)$in proving theorems in number theory. May 26 Every polynomial ring in which, the variable and coefficients belong to Z, has an infinite set of impossible prime factors. This is a consequence of a property of polynomials refered to in a recent message: If f(x) is a polynomial ring then f(x_0 + k*f(x_0)) is congruent to 0 (mod (f(x_0)). May 25 Excepting 2^n-1 all exponential functions of type a^n + c where a and n belong to N, c belongs to Z and n is not fixed, have an infinite set of odd impossible prime factors. This is a corollary of Ëuler's generalisation of Fermat's theorem - a further generalisation ( Hawaii International conference on mathematics- 2004). May 21 Background: In 1988 I read the book "one, two, three, infinity " by George Gammow. The book had a statement to the effect that no polynomial had been found such that it generates all the prime numbers and nothing but prime numbers. This was true at the time Gammow wrote the book; however subsequently a polynomial was constructed fulfiling the condition given above. I then experimented with some polynomials and found that although one cannot generally predict the prime numbers generated by a polynomial one can predict the composite numbers generated by a polynomial. Since I was originally trying to predict the primes generated by a given polynomial (which may be called "successes ") but could predict the "failures" (composite numbers) I called functions which generate failures "failure functions ". I presented this concept at the Ramanujan Mathematical society in May 1988. Subsequently I used this tool in proving a theorem similar to the Ramanujan Nagell theorem at the AMS-BENELUX meeting in 1996. Abstract definition: Let$f(x)$be a function of$x$. Then$x = g(x_0) $is a failure function if f(g(x_0)) is a failure in accordance with our definition of a failure.Note:$x_0$is a specific value of$x$. Examples: 1) Let our definition of a faiure be a composite number. Let$f(x) be a polynomial in x where x belongs to $Z$. Then $x$ = $x_0 + kf(x_0) is a failure function since these values of$x$are such that f(x) are composite. 2) Let our definition of a failure again be a composite number. Let the function be an exponential function$a^x + c where a and x belong to N, c belongs to Z $and a and c are fixed. Then$x = x_0 + k*Eulerphi(f(x_0)$is a failure function.Here also$x_0$is fixed. Here k belongs to N. 3) Let our definition of a failure be a non-Carmichael number. Let the mother function be$2^n + 49$. Then$n = 5 + 6*k$is a failure function. Here also$k$belngs to$N$. Applications: failure functions can be used for$a)$indirect primality testing and$b)\$ as a mathematical tool in proving theorems in number theory.