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## Versions

(v1) by jeremyboden 2015-05-15

## Carmichael numbers and pseudoprimes in the ring Z

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.

## Carmichael numbers and pseudoprimes in the ring Z(contd)

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 ).

## Carmichael numbers and pseudoprimes in the ring Z(contd)

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 ).

## Carmichael numbers and pseudoprimes in the ring Z(contd)

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.

## Carmichael numbers and pseudoprimes in the ring Z(contd)

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.

## Carmichael numbers and pseudoprimes in the ring Z(contd)

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.

## Impossible prime factors of exponential functions

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).

## Impossible prime factors of polynomials

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)).

## No official complaint

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.

## re: No official complaint

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.

## No official complaint

I have already mentioned it: ”failure functions”. If you search for it you will find the owner is bc1. But I am the author. However, I do not wish to make an issue of this. After all mathematics is so vast one can easily find something new- one need not misappropriate another’s intellectual property.

## re: No official complaint

The search is a bit difficult because of that the PM search function does not work nowadays. (http://planetmath.org/failurefunctions by bci1)

Jussi

## Fallout of amateur research

As I said in a recent message: mathematics is such a vast subject that something new can be discovered by even amateurs. Sometimes the discoveries may be trite. Sometimes they may be more significant. Examples: a) any prime number that ends with 1 can be a suitable base for pseudoprimality of 15. b) A corolary of Fermat’s theorem : both Mersenne primes and Mersenne composites are such that if you decrease the Mersenne prime or factors of Mersenne composites by one you get multiples of the relevant prime exponent. i.e. if 2^p-1 is prime, say P, then P-1 is a multiple of p. If 2^p-1 is composite and the factors are, say f_1, f_2… then (f_1-1) is a multiple of p and (f_2 - 1) is a multiple of p. In case this is not clear I will give numerical examples.

## Abstract algebra

There cannot be a better introduction to abstract algebra than ”Rings, fields and groups ” by RBJT Allenby. The style is very good. Incidentally there is a very simple proof of the fact that every group of order p (p is a prime number ) is cyclic. ( page 214 ).

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