I Polynomials can be used to generate a finite string of primes....

AI Thread Summary
The polynomial F(n) = n^2 - n + 41 generates prime numbers for all n less than 41. Participants discussed whether there are polynomials that can generate longer lists of primes, concluding that while some exist, the overall list of such polynomials is finite. Quadratic polynomials can generate primes, but the existence of longer lists remains uncertain. The Green-Tao theorem indicates that for any positive integer k, there is a prime arithmetic progression of length k. The discussion highlighted the complexities of prime generation through polynomials and the implications of established mathematical theorems.
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TL;DR Summary
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?
 
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(1) Yes.
(2) No.
(3) I don't think so. No for any list, yes, for longer lists.
 
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.
 
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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.

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