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

LukIn summary, the conversation discussed the polynomial F(n)=##n^2 −n+41##, which generates primes for all values of n less than 41. There are other prime-generating polynomials, but it is unknown if there is a finite or infinite list of these polynomials. The Green-Tao theorem states that for any positive integer k, there exists a prime arithmetic progression of length k, meaning there are infinitely many primes that can be generated in a sequence. This theorem was further discussed in a seminar. The conversation has now ended.
  • #1
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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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  • #2
(1) Yes.
(2) No.
(3) I don't think so. No for any list, yes, for longer lists.
 
  • #4
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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  • #5
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
 

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

What are polynomials?

Polynomials are mathematical expressions made up of variables, coefficients, and exponents. They can be used to represent a wide range of mathematical relationships, including prime numbers.

How can polynomials generate a finite string of primes?

By using specific polynomial functions, it is possible to generate a sequence of numbers that are all prime. This is known as a polynomial sequence, and it can be used to find a finite string of primes.

Is there a specific polynomial formula for generating prime numbers?

There is no single polynomial formula that can generate all prime numbers. However, there are many different polynomial functions that can generate a finite string of primes.

What are some examples of polynomial functions that can generate prime numbers?

One example is the polynomial function f(x) = x^2 + x + 41, which generates prime numbers for all values of x from 0 to 39. Another example is the polynomial function f(x) = x^2 - x + 41, which generates prime numbers for all values of x from 0 to 40.

Are there any limitations to using polynomials to generate prime numbers?

While polynomials can be used to generate a finite string of primes, they cannot generate all prime numbers. Additionally, as the values of x increase, the likelihood of the resulting number being prime decreases. Therefore, polynomials may not be a reliable method for generating large prime numbers.

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