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[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[P] failure functions by akdevaraj 6:06 am
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.

[p] Impossible prime factors of polynomials by akdevaraj 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)).

[p] Impossible prime factors of exponential functions by akdevaraj 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 √čuler's generalisation of Fermat's theorem - a further generalisation ( Hawaii International conference on mathematics- 2004).

[P] failure functions by akdevaraj 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.

[p] Carmichael numbers and pseudoprimes in the ring Z(contd) by akdevaraj May 19
Here is an algorithm that works for pseudoprimes: Let ab be a composite number where a and b are prime. Then ab + 1 is a base for pseudoprimality of ab.

[p] Carmichael numbers and pseudoprimes in the ring Z(contd) by akdevaraj May 19
Here is an algorithm that works for pseudoprimes: Let ab be a composite number where a and b are prime. Then ab + 1 is a base for pseudoprimality of ab.