On primitive integer solutions of the Diophantine equation and related results

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• 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=47c6596ae4f6fee65fc372232353b70b">class="imgLazyJSB inlineImage" height="18" width="81" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002450-si2.gif">class="mathContainer hidden">class="mathCode">$T^2=G(\bar{X})$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=7b09602c5336e67810acd1b3b94d6b2d">class="imgLazyJSB inlineImage" height="18" width="132" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002450-si3.gif">class="mathContainer hidden">class="mathCode">$\bar{X}=(X_1,\dots ,X_m)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si4.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=5b8841dd6cf489f917198b84e099530d" title="Click to view the MathML source">m=3class="mathContainer hidden">class="mathCode">$m=3$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si5.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=ab8d36fe8fe36fae90f36515d7f6a137" title="Click to view the MathML source">m=4class="mathContainer hidden">class="mathCode">$m=4$ and G   is a specific homogeneous quintic form. First, we prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si6.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=2da86d03b0f08243b70c486ab962d951" title="Click to view the MathML source">F(x,y,z)=x²+y²+az²+bxy+cyz+dxz∈Z[x,y,z]class="mathContainer hidden">class="mathCode">$F(x,y,z)=x^2+y^2+a{z}^2+bxy+cyz+dxz\in \mathbb{Z}[x,y,z]$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si164.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=5c5ad56b194608e73919f7d43494e145" title="Click to view the MathML source">(b−2,4a−d²,d)≠(0,0,0)class="mathContainer hidden">class="mathCode">$(b-2,4a-d^2,d)\ne (0,0,0)$, then for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si133.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=4ce0208c711ff5ba80cf764b6f1b62dc" title="Click to view the MathML source">n∈Z鈭杮0}class="mathContainer hidden">class="mathCode"> the Diophantine equation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si9.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=ead3ba96051828d96d12d0c060cd99c3" title="Click to view the MathML source">t²=nxyzF(x,y,z)class="mathContainer hidden">class="mathCode">$t^2=nxyzF(x,y,z)$ has a solution in polynomials x, y, z, t   with integer coefficients, with no polynomial common factor of positive degree. In case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si10.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=dcda36abd583123af4421a592f64ec2d" title="Click to view the MathML source">a=d=0class="mathContainer hidden">class="mathCode">$a=d=0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si11.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=410df7a47b302272b461a8409aa03f22" title="Click to view the MathML source">b=2class="mathContainer hidden">class="mathCode">$b=2$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si12.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=4b67652db3121dfeb6ebd527d50e1888" title="Click to view the MathML source">n∈Q鈭杮0}class="mathContainer hidden">class="mathCode"> 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si14.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=a43e55141c50c55ceb6020b2cf3d72ac">class="imgLazyJSB inlineImage" height="18" width="219" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002450-si14.gif">class="mathContainer hidden">class="mathCode">$T^2=a{X}_1^5+b{X}_2^5+c{X}_3^5+d{X}_4^5$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si104.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=0c7933ccd49c2d84c103bd91133ecfe5" title="Click to view the MathML source">a,b,c,d∈Zclass="mathContainer hidden">class="mathCode">$a,b,c,d\in \mathbb{Z}$. In particular, we prove that for each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si16.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=cf1d90565dc2c0e7f2cc7a7eb473571f" title="Click to view the MathML source">m,n∈Z鈭杮0}class="mathContainer hidden">class="mathCode">, the Diophantine equation has a solution in polynomials which are co-prime over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si18.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=8c5124df2c311ba45db98c4afa25cf92" title="Click to view the MathML source">Z[t]class="mathContainer hidden">class="mathCode">$\mathbb{Z}[t]$. Moreover, we show how a modification of the presented method can be used in order to prove that for each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si12.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=4b67652db3121dfeb6ebd527d50e1888" title="Click to view the MathML source">n∈Q鈭杮0}class="mathContainer hidden">class="mathCode">, the Diophantine equation has a solution in polynomials which are co-prime over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002450&_mathId=si18.gif&_user=111111111&_pii=S0022314X15002450&_rdoc=1&_issn=0022314X&md5=8c5124df2c311ba45db98c4afa25cf92" title="Click to view the MathML source">Z[t]class="mathContainer hidden">class="mathCode">$\mathbb{Z}[t]$.