Just to clarify, I believe that Q.is_locally_represented_number(42), where Q = DiagonalQuadraticForm(ZZ, [1,4,4]) should return False.
An integer m is locally represented if m is represented by Q mod p ^a for every prime p and every non-negative integer a and also that it is represented over the real numbers. This is a paraphrase from Jonathan Hanke's article "Local Densities and Explicit Bounds for Representability by a Quadratic Form." By m is represented by Q, I mean that there exists a vector v with integer entries such that Q(v) = m. What I was saying in my post is that if you take the quadratic form x^2 + 4*y^2 + 4*z^2 (mod 2^2), you get that this expression is equivalent to x^2 (mod 4). However, 42 is equivalent to 2 (mod 4), and no number equivalent to 2 (mod 4) is a square. You could do a brute force calculation to verify this fact. By the way, if the following actually occurs with 6.3.beta5, the program that I'm writing in Sage is going to be severely inaccurate in future versions of Sage unless this bug is fixed: sage: Q = DiagonalQuadraticForm(ZZ,[1,4,4]) sage: Q.is_locally_represented_number(42) True On Tuesday, July 1, 2014 2:03:35 PM UTC-4, Dominique Laurain wrote: > > Please check my answer below...I am newbie in these subjects... > > The point of Edna's Jones question is about "locally", no ? > > As Pete L.Clark points out in http://www.math.uga.edu/~pete/*CasselsLemma* > .pdf > > a number n can be "locally" (in Qp) represented but not integrally > represented....read some examples in Clark's paper > > The ZZ field is forced in DiagonalQuadraticForm() first argument to > specify integer quadratic forms (with integer coefficients). > > It doesn't matter if you use ZZ or python int for > locally_represented_number() argument (+1 : previous answer). > > You have Gauss's theorem for representing number n different of > 4^k(8l+7) by the form x^2 + y^2 + z^2 with x,y,z in Q > and 42 is represented by it, so it is represented by the equivalent form > x^2 + (2y)^2 + (2z)^2. > > That's why function locally_represented returns true ... while obviously > no integers x,y,z can be set to have x^2 + 4y^2 + 4z^2 = 42 > > Dominique > > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
