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The classical result by Dirichlet proved by Schering and Weber (independently) is that, every primitive quadratic form with with discriminant different perfect square represents infinitely many primes.

AS we know some irreducible cubic equation, quadratic equations are capable of representing prime numbers, are there any similar result on function fields?

i.e can we think of "some special equations " generating irreducibles in function fields

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  • $\begingroup$ In what sense are you generating prime numbers from quadratic equations? $\endgroup$ – Hurkyl Dec 8 '17 at 5:17
  • $\begingroup$ I am asking about the analogous result for function fields $\endgroup$ – thanks Dec 8 '17 at 5:31
  • $\begingroup$ I don't know what you mean even in regard to the integers. $\endgroup$ – Hurkyl Dec 8 '17 at 5:36
  • $\begingroup$ In particular, the first thing that springs to mind is the story "Look at the values of $x^2 + x + 41$. It looks like it generates primes, doesn't it? But surprise! It's just a coincidence that the first 40 values are prime; its 41st is not prime, and in fact there's an easy proof that polynomials can't be prime generating." $\endgroup$ – Hurkyl Dec 8 '17 at 5:45
  • $\begingroup$ The solution to Diophantine equations is the parameterization. Choose as a parameter to any function. Will be obtained identities with different shape. So some identities of Ramanujan found. $\endgroup$ – individ Dec 8 '17 at 6:24

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