Re: [math-fun] A few questions about prime generating functions
That Wikipedia article Formulas for primes contains this interesting tidbit: ----- It is not even known whether there exists a univariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; ----- The article appears to be referring to *integer* polynomials. What about a *real* polynomial P(x), deg(P) >= 2 ??? Can some such P(x) be proved to represent infinitely many primes? —Dan ----- II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? -----
Hello, I have found one formula, not a polynomial, a family of formulas with real coefficients, here : http://plouffe.fr/NEW/ and here ; 100 primes in a row here : http://plouffe.fr/Record%20100%20primes%20sequence.txt article on ArXiv : https://arxiv.org/ftp/arxiv/papers/1901/1901.01849.pdf It is (partly) based on Mills and Wright formulas for primes. Best regards, Simon Plouffe Le 2019-04-29 à 20:36, Dan Asimov a écrit :
That Wikipedia article Formulas for primes contains this interesting tidbit:
----- It is not even known whether there exists a univariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; -----
The article appears to be referring to *integer* polynomials. What about a *real* polynomial P(x), deg(P) >= 2 ???
Can some such P(x) be proved to represent infinitely many primes?
—Dan
----- II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image:
|f(Z+) ∩ Primes| = oo
??? -----
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Dan Asimov -
Simon Plouffe