User randomvisitor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:21:22Z http://mathoverflow.net/feeds/user/15559 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66796/a-question-on-m-mignottes-paper-pethos-cubics A question on M. Mignotte's Paper: "Petho's Cubics" RandomVisitor 2011-06-03T04:27:39Z 2011-06-03T04:27:39Z <p>I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in <strong>Publ. Math. Debrecen, 56/3-4 (2000)</strong>) unfortunately I don't understand section 4 of the paper. I am hoping someone could help me out here. </p> <p>In this paper the author considers a family of cubics: $$\Phi_n (x,y) = x^3 - nx^2y - (n+1)xy^2 - y^3$$ in section 4 (at the very end of the paper) the author uses the following Lemma:</p> <blockquote> <p><strong>Lemma.</strong> Let $\Lambda = u \alpha + v \beta + \gamma$, where $\alpha, \beta$ and $\gamma$ are non-zero real numbers and where $u$ and $v$ are rational integers, with $|u| \leqslant A$. Let $Q>0$ be a real number. Suppose that $\theta_1$ and $\theta_2$ satisfy: $$|\theta_1 - \tfrac{\alpha}{\beta} | &lt; \frac{1}{100Q^2}, \qquad \text{and} \qquad |\theta_2 - \tfrac{\gamma}{\beta} | &lt; \frac{1}{Q^2}.$$ Let $\tfrac{p}{q}$ be a rational number with $1 \leqslant q \leqslant Q$ and $|\theta_1 - \tfrac{p}{q} |&lt; \tfrac{1}{q^2}$ and suppose that $q\|q\theta_2\| \geqslant 1.01A + 2$, then: $$|\Lambda| \geqslant \frac{|\beta|}{Q^2}.$$</p> </blockquote> <p>in order to show that for small values of $n$ with $n \geqslant 10$ the cubic has no non-trivial solutions. What I don't understand is that how this Lemma could be applied to this case. </p> <p>My own sense is that $(p,q)$ is an integer solution for $\Phi_n (x,y) = 1$, that $\theta_i$ are roots of the $\Phi_n (x,1)=0$ and $\Lambda$ is a linear form in logarithm corresponding to $\Phi_n (x,y) = 1$. However I don't see how $\theta_i$ are related to $\tfrac{\alpha}{\beta}$ and $\tfrac{\beta}{\gamma}$. Because $\alpha, \beta$ and $\gamma$ are logarithms. </p>