Polynomials generating consecutive primes: longest sequences and finiteness

[mathman]

TL;DR

Polynomials can be used to generate a finite string of primes

F(n)=$n^2 −n+41$ generates primes for all n<41.

Questions:
(1) Are there polynomials that have longer lists?

(2) Is such a list of polynomials finite (yes, no, unknown)?

(3) Same questions for quadratic polynomials?

 

[fresh_42]

(1) Yes.
(2) No.
(3) I don't think so. No for any list, yes, for longer lists.

 

[malawi_glenn]

(1) here are some prime number generating polynomials https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

 

[TeethWhitener]

It doesn't even have to be quadratic: the Green-Tao theorem states that for any positive integer $k$, there exists a prime arithmetic progression of length $k$. In other words, for any $k$, there exists a prime $p$ and a positive integer $n$ which generates the sequence $\{p, p+n, p+2n, \dots p+(k-1)n\}$ where all the members of the sequence are prime.

 

[jedishrfu]

To expand on @TeethWhitener mention of the Green-Tao Theorem, here is a seminar on the theorem:



Since this thread has run its course, it's time to close it and thank everyone who contributed here.

Jedi