representation of integers as the product of linear forms in three variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:46:30Z http://mathoverflow.net/feeds/question/67123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67123/representation-of-integers-as-the-product-of-linear-forms-in-three-variables representation of integers as the product of linear forms in three variables Tom Hunt 2011-06-07T12:39:10Z 2011-06-07T15:43:11Z <p>I would like to find all integer triples (x,y,z) such that: $\prod_{\theta}(x + y \theta + z \theta^2)=1$, where $\theta$ runs through the solutions to the cubic $x^3 + x^2 - 2x - 1=0$. </p> <p>In his book "Diophantine equations" (p. 111-12) Mordell gives the equation </p> <p>$w^n = \prod_{\theta}(x + y \theta + z \theta^2)$ </p> <p>where $\theta=\theta_1, \theta_2, \theta_3$ are the solutions to a cubic equation with integer coefficients. </p> <p>Mordell's partial solution is,</p> <p>$x + y\theta_1 + z \theta_1^2 = (p + q\theta_1 + r\theta_1^2)^n$,</p> <p>$x + y\theta_2 + z \theta_2^2 = (p + q\theta_2 + r\theta_2^2)^n$,</p> <p>$x + y\theta_3 + z \theta_3^2 = (p + q\theta_3 + r\theta_3^2)^n$,</p> <p>$w = \prod_{\theta}(p + q \theta + r \theta^2)$</p> <p>where $p,q,r$ are arbitrary integers and $n$ runs through the integers.''</p> <p>He continues to say, " the general solution depends upon the theory of algebraic numbers and is connected with the units in an algebraic number field". </p> http://mathoverflow.net/questions/67123/representation-of-integers-as-the-product-of-linear-forms-in-three-variables/67148#67148 Answer by David Loeffler for representation of integers as the product of linear forms in three variables David Loeffler 2011-06-07T15:43:11Z 2011-06-07T15:43:11Z <p>This isn't really a research-level question, and hence belongs more on math.SE than here, but here goes anyway...</p> <p>Let $K$ be the number field $\mathbb{Q}(\theta)$ where $\theta$ is a root of your cubic $f$. Then the ring of integers <code>$\mathcal{O}_K$</code> of $K$ is <code>$\mathbb{Z}(\theta)$</code>, and we are reduced to looking for elements <code>$\omega \in \mathcal{O}_K^\times$</code> such that $\operatorname{Norm}_{K/\mathbb{Q}}(\omega) = 1$. It takes about 0.01 seconds for my computer to tell me that $\mathcal{O}_K^\times$ is isomorphic to $\mathbb{Z}^2 \times \mathbb{Z}/2$, and the index 2 subgroup of units of norm 1 is isomorphic to $\mathbb{Z}^2$, with independent generators $\theta-1$ and $-\theta-1$.</p> <p>So the general solution is $$x + y \theta + z \theta^2 = (\theta - 1)^a (-\theta-1)^b$$ for any $a, b \in \mathbb{Z}$.</p> <p>(If you're unfamiliar with the theory of algebraic number fields I'm using here, Stewart and Tall's book is a good reference.)</p>