Timeline for Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13 at 1:04 | comment | added | HGF | @WillJagy Lemmermeyer in this MO mathoverflow.net/questions/144544/… said that“Meyer (Über einen Satz von Dirichlet, Crelle 103 (1888)) proved that a primitive binary quadratic form with nonsquare discriminant represents infinitely many primes that lie in any given compatible residue class modulo a given integer N”。 Unfortunately, I don't know German。 | |
| Sep 13 at 1:02 | comment | added | HGF | @WillJagy Blair K. Spearman and Kenneth S. Williams shown that every such arithmetic progression either contains no primes of the form $x^2 + 14y^2$ or it contains primes of both forms$ x^2 + 14y^2$ and $2x^2 + 7y^2$ in the paper: Representing Primes by Binary Quadratic Forms,The American Mathematical Monthly , May, 1992, Vol. 99, No. 5 (May, 1992), pp. 423-426. I also think the both represent infinitely many prime numbers in each congruent class but I don't know how to prove. | |
| Sep 12 at 18:03 | comment | added | Will Jagy | Yes; in the first edition, Theorem 9.12 on page 188 that the Dirichlet density of represented primes exists. Further more, if the form is "ambiguous," such as your $x^2 + 14 y^2$ or your $2 x^2 + 7 y^2,$ the density is $\frac{1}{2 h(D)}.$ Let's see, you also want primes in a specific arithmetic progression. The answer is yes, but I'm not sure of a place in this book. | |
| Sep 12 at 7:40 | history | edited | HGF | CC BY-SA 4.0 |
added 351 characters in body
|
| Sep 12 at 7:30 | history | asked | HGF | CC BY-SA 4.0 |