Figurative Numbers And Repdigits In The Recursive Sequence

The core question of Number Theory is the expressions of integers.Fibonacci numbers are the most important and fundamental sequences in second-order linear recursive sequences,and their properties have always been one of the important research contents in Number Theory.The Narayana cow’s sequence is also a representative sequence of third-order recursive sequences.In this thesis,it is mainly solved that the Fibonacci sequences which are related to the binomial coefficients,i.e.the equation F_n=1/2x(x-1)+1,and which repdigit could be written as the difference of two Narayana cow’s numbers.For Fibonacci numbers of the form (?)+1,which is equivalence to 8F_n=(2x-1)~2+7,using the periodicity of Fibonacci numbers with respect to the remaining classes of modulos,solving the necessary conditions satisfied by the square numbers in the sequence by stepwise modulometry,and further solving for all the above conditions by establishing the identity of Jacobian symbols,proving that there are only finite Fibonacci numbers that can be tabled as 1/2x(x-1)+1.For the repdigit which could be written as the difference between two Narayana cow’s numbers,firstly,we given a large upper bound is initially obtained by the linear form in logarithms and the properties of Narayana cow’s sequences.Then,a computable upper bound is obtained by a further reduction method.Finally,all solutions are calculated by Mathematica mathematical software in the obtained range.