Study On The Solutions Of Diophantine Equations And Inequalities Related To Pell Numbers

Let {Pn}n≥0 and {Pn}n≥0 be the Pell and the Padovan sequences given by the initial values P0=0,P1=1,P0=0,P1=P2=1 and the recurrence formulas Pn+2=2Pn+1+Pn,Pn+3=Pn+1+Pn for all n≥0,respectively.Let(?)is the concatenation of three repdigits in base 10 with c1,c2,c3∈{0,1,…,9},c1>0,c1≠c2,c2≠c3,mi∈Z+(i=1,2,3).By using the linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method,in this paper we find all the positive integer solutions of the Diophantine equation(?)are(n,Pn)∈{(7,169),(8,408),(9,985).We prove all the positive integer solutions of the Diophantine inequality |Pn-Pt|<Pt1/2 are(n,t)∈{(1,1),(1,2),(2,1),(2,2),(3,1),(3,2),(4,2),(5,2),(6,2),(6,3),(7,3),(8,3),(9,3),(10,4),(11,4),(14,5),(17,6)},and it is also proved that Diophantine inequality |Pn+Pm-Pt|<Pt1/2 has 117 groups of positive integer solutions(n,m,t)when n≥m,where n≤58,m≤38 and t≤19.The specific results can be see in theorem 1.3.