[math-fun] A few questions about prime generating functions
I. It's widely believed that there's no nice function f : Z+ —> Z+ whose values are all prime: f(Z+) ⊂ Primes , where Primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...}. Are there any actual theorems that say as much — necessarily defining "nice" ? Or at least restricting the classes of nice functions for which this might be possible? How about heuristics indicating why this should be hard? * * * II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? Dirichlet's famous 1837 theorem states that every arithmetic sequence of form {X_n = A + B n}, where A, B are relatively prime integers, contains infinitely many primes.* But I don't know of any other cases. Is that because they aren't known? —Dan ————— * I once read the proof (as a very readable chapter in Elliptic Curves by Anthony Knapp) and thought it astonishing innovative, but especially for that long ago!
On Sun, Apr 28, 2019 at 9:28 PM Dan Asimov <dasimov@earthlink.net> wrote:
II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image:
|f(Z+) ∩ Primes| = oo
???
f(z) = z and f(z) = 2z+1 come to mind... -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
The interesting question would be to find a function for which almost all f(Z+) is prime; are there any simple functions for which the asymptotic density of primes in f(Z+) is a larger order of magnitude than 1/log(n)? On Sun, Apr 28, 2019 at 10:32 PM Mike Stay <metaweta@gmail.com> wrote:
On Sun, Apr 28, 2019 at 9:28 PM Dan Asimov <dasimov@earthlink.net> wrote:
II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image:
|f(Z+) ∩ Primes| = oo
???
f(z) = z and f(z) = 2z+1 come to mind... -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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Some relevant background material here: https://en.wikipedia.org/wiki/Formula_for_primes On Sun, Apr 28, 2019 at 9:28 PM Dan Asimov <dasimov@earthlink.net> wrote:
I. It's widely believed that there's no nice function f : Z+ —> Z+ whose values are all prime:
f(Z+) ⊂ Primes
, where
Primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...}.
Are there any actual theorems that say as much — necessarily defining "nice" ?
Or at least restricting the classes of nice functions for which this might be possible?
How about heuristics indicating why this should be hard?
* * *
II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image:
|f(Z+) ∩ Primes| = oo
???
Dirichlet's famous 1837 theorem states that every arithmetic sequence of form
{X_n = A + B n},
where A, B are relatively prime integers, contains infinitely many primes.*
But I don't know of any other cases. Is that because they aren't known?
—Dan ————— * I once read the proof (as a very readable chapter in Elliptic Curves by Anthony Knapp) and thought it astonishing innovative, but especially for that long ago!
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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Dan Asimov -
Michael Collins -
Mike Stay