the multiplicativ structur of the discriminant for quadratic polynomials

[bhelmes]

A peaceful day for all,

there is a nice table for the discriminant of special quadratic polynomials and their resulting pattern:

http://devalco.de/periodic_system_of...riminants.html

It is a preview and i think their should be some more information added.

Nevertheless for those mathematicans having fun with primes
i try to build up a periodical system which seems to have some relationship with the periodical system of chemical elements.

I have calculated the polynomial f(n)=n²+bn+c in some limits for b and c,
chosen only the polynomials which construct some infinite series of prime numbers, and tried to get a relationship for those discriminants which
consists of two primes and the discriminants with one prime.

Mathematical feedback and suggestion for improvements
are welcome

Greetings from the primes
Bernhard

[science_man_88]

Originally Posted by bhelmes

A peaceful day for all,

there is a nice table for the discriminant of special quadratic polynomials and their resulting pattern:

http://devalco.de/periodic_system_of...riminants.html

It is a preview and i think their should be some more information added.

Nevertheless for those mathematicans having fun with primes
i try to build up a periodical system which seems to have some relationship with the periodical system of chemical elements.

I have calculated the polynomial f(n)=n²+bn+c in some limits for b and c,
chosen only the polynomials which construct some infinite series of prime numbers, and tried to get a relationship for those discriminants which
consists of two primes and the discriminants with one prime.

Mathematical feedback and suggestion for improvements
are welcome

Greetings from the primes
Bernhard

if you say discriminant = b^2-4c for the first one you get 16-4*(-11) = 16+44=60 not 15. the only reason you can reduce the discriminant to b^2-4c is because a=1 ( aka your polynomials are all monic). in fact any time b is 4 you seem to be reducing it by a factor of 4. edit: and some of these could be changed to other polynomials to compress the n values for the primes for example n²+4n-11 is only odd when n is even in which case you get 4y^2+8y-11. this polynomial will create the same primes as the previous but with fewer values needing to be tested overall ( on average half as many).

[bhelmes]

A peaceful evening for all,

there is a collection of monic quadratic irreducible polynomials f(n)=n^2+bn+c where the discriminant is b^2-4c.

I choose the polynomials with discriminant=p1*p2, where p1 and p2 are primes and add the polynomials with discriminant p1 and p2 :

http://devalco.de/triple_system.php

It seems to be that the three polynomials with discr. = p1*p2, p1 and p2
"contains" all primes.

If someone has a good mathematical proof, it would be nice to get a description.

Have a lot of fun with primes
Bernhard

[MattcAnderson]

Hi Bernhard and everybody,

I appreciate the work you did on your webpage Bernhard. I was able to scroll down and choose a polynomial and then see the sequence when the input is an integer.

Keep up the good work.

Regards,
Matt