[P] **failure functions ** by akdevaraj May 21Background: 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 19Here 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 19Here 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 19Here 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 18Can we predict the bases for psedoprimes in Z? To some extent
we can. For example any prime number which ends with one seems to
be a base suitable for the pseudoprime 15. (to be continued ).

[p] **Carmichael numbers and pseudoprimes in the ring Z(contd)** by akdevaraj May 18Can we predict the bases for psedoprimes in Z? To some extent
we can. For example any prime number which ends with one seems to
be a base suitable for the pseudoprime 15. (to be continued ).

[p] **Carmichael numbers and pseudoprimes in the ring Z** by akdevaraj May 17AS is well known Carmichael numbers are pseudo to any base in Z
not coprime with the number under consideration. In the case
of pseudoprimes in Z how to find a base such that the
composite number is pseudo to that base? Fortunately we can
run the following program in pari to find such bases:
Let us take the simple example of 15. Then {p(n)=(n^14-1)/15;
for (n=1,12,print (p(n))). I find that 4 is the first such base.

**The search feature...** by jeremyboden May 15>

[p] **A property of polyomials (concluding).** by akdevaraj May 15Let f(x) be a monic polynomial. Then a) f(x_0 +k*f(x_0)) is
congruent to 0 mod (f(x_0)) b) it is also congruent to 0 mod(f(x_0+1)
c) It is also congruent to 0 mod(x_0 + i) in the case when
x is a Gaussian integer and d) it is also congruent to 0
mod (f(x_0) and mod(f(x_0 + 1) when x is a square matrix in which
each element belongs to Z(i). Here k belongs to N.

[p] **A property of polynomials (contd)** by akdevaraj May 14This property is exhibited by monic polynomials in Z
where the variable is a square matrix in which each element belongs
to Z.

[p] **A property of polyomials (contd)** by akdevaraj May 13Apparently this property is exhibited by monic polynomials in
Z(i) too.

[p] **A property of polyomials** by akdevaraj May 13Let us, for the present, confine ourselves to the ring of integers.
Sometime ago I had stated that the following property of
polynomial holds: let f(x) be a polynomial ring,where x belongs to Z. Let x_0 be a specific
value of x.. Then f(x_0 + k*f(x_0)) is
congruent to 0 (mod f(x_0). Here k belongs to N. Now it
looks as if f(x_0 + k*f(x_0)) is also congruent to f(x_0+1), in the case of monic polynomials
only.

[p] **Fermat's theorem in Z(i) (contd)** by akdevaraj May 8Program in pari for generating Gaussian integer quotients - an
example: {p(n) = ((n + I)^12-1)/13}. This, of course, fails when n
is such that (n + I) is not coprime with the prime factors of
13 in Z(i).

[p] **Math special characters** by pahio May 6Hi unlord,
I have found a partial solution to the problem concerning the math
fraktur letters in the PM articles: Remove from the LaTeX-form text
the expressions containing thmplain, PMlinkescapeword, PMlinkname.
Then the fraktur letters are visible. The same concers probably the
math calligraphy letters.
A new(?) problem may be that the autolinking works nowadays quite
poorly.
Is there any knowledge on the final great rebuilding of PlanetMath?
Many persons are waiting it.
Regards,
Jussi