# What is the maximum variety of tops created back to back created by a polynomial of level $a$?

Allow $p(n)$ be a polynomial of level $a$. Start of with diving in debates from absolutely no and also rise one integer at the time. Take place till you have actually come with an integer argument $n$ of which $p(n)$ is value is not prime and also count the variety of distinctive tops your polynomial has actually created.

Inquiry : what is the maximum variety of distinctive tops a polynomial of level $a$ can create by the procedure defined over? In addition, what is the basic kind of such a polynomial $p(n)$?

This inquiry was motivated by this article.

Many thanks,

Max

[Please keep in mind that your polynomial does not require to create successive tops, just tops at successive favorable integer debates. ]

Here is outcome by Rabinowitsch for square polynomials.

$n^2+n+A$ is prime for $n=0,1,2,...,A-2$ if and also just if $d=1-4A$ is squarefree and also the class variety of $\mathbb{Q}[\sqrt{d}]$ is $1$.

See this write-up for information.

http://matwbn.icm.edu.pl/ksiazki/aa/aa89/aa8911.pdf

Also below is a checklist of fictional square areas with class number $1$ http://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one#Imaginary_quadratic_fields

There are several various other write-ups concerning prime getting (square) polynomials that you can google.

The Green - Tao Theorem mentions that there are randomly lengthy math developments of tops ; that is, series of tops of the kind $$ b , b+a, b+2a, b+3a,... ,b+na $$ Since such a development will certainly be the first $n$ values of the polynomial $ax+b$, this indicates that also for level 1, there is no upper bound to the amount of tops straight a polynomial can create.

Here is a relevant reality which could additionally be of passion. There exists a polynomial in 26 variables with the building that, if you connect in integers for all 26 variables, and also the result is a favorable number (it will certainly constantly be an integer), after that the result is prime. The polynomial can be located below :

http://en.wikipedia.org/wiki/Formula_for_primes

Furthermore, every prime takes place as a result of this polynomial. This is a really indirect means to create tops, due to the fact that there is no very easy means to recognize if an offered input will certainly offer a favorable result.

There is evidently an additional polynomial in 10 variables that does this, yet Wikipedia does not clearly write it.