On primitive integer solutions of the Diophantine equation and related results

详细信息    查看全文

• 作者：Maciej Gawron ^{maciekggawron@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Maciej Ulas ; ^{maciej.ulas@uj.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11D41
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：159
• 期：Complete
• 页码：101-122
• 全文大小：431 K

文摘

In this paper we investigate Diophantine equations of the form $T^2=G(\bar{X})$, $\bar{X}=(X_1,\dots ,X_m)$, where m=3$m=3$ or m=4$m=4$ and G   is a specific homogeneous quintic form. First, we prove that if F(x,y,z)=x²+y²+az²+bxy+cyz+dxz∈Z[x,y,z]$F(x,y,z)=x^2+y^2+a{z}^2+bxy+cyz+dxz\in \mathbb{Z}[x,y,z]$ and (b−2,4a−d²,d)≠(0,0,0)$(b-2,4a-d^2,d)\ne (0,0,0)$, then for all n∈Z鈭杮0} the Diophantine equation t²=nxyzF(x,y,z)$t^2=nxyzF(x,y,z)$ has a solution in polynomials x, y, z, t   with integer coefficients, with no polynomial common factor of positive degree. In case a=d=0$a=d=0$, b=2$b=2$ we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideration. As an application of our result we prove that for each n∈Q鈭杮0} the Diophantine equation has a solution in co-prime (non-homogeneous) polynomials in two variables with integer coefficients. We also present a method which sometimes allows us to prove the existence of primitive integer solutions of more general quintic Diophantine equation of the form $T^2=a{X}_1^5+b{X}_2^5+c{X}_3^5+d{X}_4^5$, where a,b,c,d∈Z$a,b,c,d\in \mathbb{Z}$. In particular, we prove that for each m,n∈Z鈭杮0}, the Diophantine equation has a solution in polynomials which are co-prime over Z[t]$\mathbb{Z}[t]$. Moreover, we show how a modification of the presented method can be used in order to prove that for each n∈Q鈭杮0}, the Diophantine equation has a solution in polynomials which are co-prime over Z[t]$\mathbb{Z}[t]$.