Research On The Solutions Of Diophantine Equations And Inequalities Related To Balancing Numbers

Let {B_n}_n≥0 and {N_n}_n≥0 be the Balancing and the Narayana’s cow sequences given by the initial conditions B₀=0,B₁=1,N₀=0,N₁=N₂=N₃=1 and the recurrence formulas B_n+1=6B_n-B_n-1（n≥1）N_n=N_n-1+N_n-3（n≥3）,respectively.In this paper,we proved that Balancing numbers which are concatenations of three repdigits,and Narayana numbers and the products of two Narayana numbers which are close to Balancing numbers by using the linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction methods.Let Z⁺ be a positive integer set,and the following results are obtained:1.Let a₁,s₁,s₂,s₃∈Z⁺,a₁,a₂,a₃={0,1,…,9}.a₁≠a₂ and a₂≠a₃,then Diophantine equation（?） has positive integer solutions（n,B_n）∈{（4,204）,（5,1189）}.2.Let n,k∈ Z⁺,then Diophantine inequality |N_n-B_k|<B_k^1/2 has positive integer solutions（n,k）∈{（1,1）,（2,1）,（3,1）,（6,2）,（7,2）}.3.Let n,m,k ∈Z⁺ and 4 ≤m≤n,then Diophantine inequality|N_nN_m-B_k|<B_k^1/2 has positive integer solutions（n,m,k）∈ {（4,4,2）,（5,4,2）,（6,4,2）,（7,7,3）,（8,6,3）,（9,5,3）,（10,4,3）,（15,8,5）,（18,5,5）,（19,4,5）}.