Multiplicative Diophantine equations with factors from different Lucas sequences

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• 作者：Carlos Alexis Gó ; mez Ruiz^a ; ^{carlos.a.gomez@correounivalle.edu.co" class="auth_mail" title="E-mail the corresponding author} ; Florian Luca^b ; ^c ; ^{florian.luca@wits.ac.za" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11B39 ; 11D61 ; 11J86
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：170
• 期：Complete
• 页码：282-301
• 全文大小：386 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301573&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301573&_rdoc=1&_issn=0022314X&md5=1cf10286596832308b92cf613741c0e0" title="Click to view the MathML source">k≥2class="mathContainer hidden">class="mathCode">$k\ge 2$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301573&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301573&_rdoc=1&_issn=0022314X&md5=79721186b8304a2df9b19eb0b64188a2">class="imgLazyJSB inlineImage" height="21" width="173" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301573-si2.gif">class="mathContainer hidden">class="mathCode">${\{u_n^{(1)}\}}_{n\ge 0},\dots ,{\{u_n^{(k)}\}}_{n\ge 0}$ be k different nondegenerate binary recurrent sequences of integers. In this paper, we show that under certain conditions, there are only finitely many of k  -tuples of the form class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301573&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301573&_rdoc=1&_issn=0022314X&md5=d97d5910920f214304c8aa702ea03c29">class="imgLazyJSB inlineImage" height="21" width="102" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301573-si3.gif">class="mathContainer hidden">class="mathCode">$(u_{n_1}^{(1)},\dots ,u_{n_k}^{(k)})$ whose contents are multiplicatively dependent. Furthermore, we determine all such instances when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301573&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301573&_rdoc=1&_issn=0022314X&md5=2565a394714f57a7e336bb92a46e2eca" title="Click to view the MathML source">k=3class="mathContainer hidden">class="mathCode">$k=3$ and the three sequences are the sequence of Fibonacci numbers, Pell numbers, and base 10 repunits, respectively.