Some data: 100% of integers are represented by the form XY. 33% of integers are represented by the form 3XY. 0% are represented by the forms X^2+Y^2 and X^2-2Y^2. The last line is a bit delicate: If we restrict gcd(X,Y)=1, the forms represent numbers with no 4K-1 divisor and no 8K+-3 divisor, which slowly -> 0% for large numbers. If I've done my estimation right, the addon for allowing gcd(X,Y)>1 doesn't change things. The fact that X^2+Y^2 represents only non-negatives, while X^2-2Y^2 represents both signs, should be mentioned. But 2*0% = 0%, so it doesn't matter. Rich ------ Quoting Neil Sloane <njasloane@gmail.com>:
The Cebotarev density theorem is relevant here, although it does not do exactly what you ask for - see Cox,. Primes of Form x^2+ny^2 (Wiley)
On Tue, Oct 7, 2014 at 4:25 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Given arbitrary integers A, B, C they determine the quadratic form
Q(x,y) = Ax^2 + Bxy + Cy^2 .
* Does the limiting density dens(A,B,C) of the set of integers represented by Q always exist???
* If so, can one determine its value???
Here dens(A,B,C) is defined as follows:
Define the set
Q_R := {Q(x,y) | (x,y) \in Z^2 with x^2+y^2 <= R^2} . Note that any repeated value of Q(x,y) appears in Q_R with multiplicity = 1 here.
Now let
dens(A,B,C) := limit as R -> oo of (card(Q_R) / (pi R^2)),
if it exists. (Here pi R^2 is a stand-in for the number of lattice points lying inside the disk of radius R about the origin. But the two expressions are asymptotic to each other, so this should not be a problem.)
Problem: Given any integers A, B, C does the limit dens(A,B,C) exist? And if so, what is its value? -------- -------------------------------------------------------------------------------------------
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun