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

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

Yes, well deserved Happy Birthday. which makes me think of a sequence for the occasion 1, 75, 900, 3900, 27393, 250000, 657450, 39447000, 2366000000, ... Any clue ? 1 Neil James Alexander Sloane 75 years, 900 months, 3900 weeks, 27394 days, 250000 sequences, 657450 hours, 39447000 minutes, 2366000000 seconds, approximately on October 10 2014. Happy Birhday Neil, Simon 2014-10-07 12:45 GMT+02:00 James Propp <jamespropp@gmail.com>:

Are any of you involved in running the DIMACS meeting and associated events, or involved in running OEIS or seqfans?

I won't be able to attend the meeting, but the Handbook and the website have played a huge role in my career over the last 40 years, and if there's an opportunity for me to express my gratitude to Neil, perhaps in the form of a toast to be delivered in absentia at the conference dinner, or a short article in a festschrift, or anything like that, I'd really like to know!

(I've wrote to Eugene Fiorini at DIMACS last Friday and the Friday before that, but have received no reply.)

Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun