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## Latest Messages

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.

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.

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.

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.

May 18
Can 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 ).

May 18
Can 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 ).

May 17
AS 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.

May 15
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May 15
Let 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.

May 14
This property is exhibited by monic polynomials in Z where the variable is a square matrix in which each element belongs to Z.

May 13
Apparently this property is exhibited by monic polynomials in Z(i) too.

May 13
Let 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.

May 8
Program 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).

May 6
Hi 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