Research On The Representations Of Lucas Sequences

The diophantine equation is one of the number theories,which considers an equation(or equations)where the number of variables is larger than the number of equation(s).Scholars have done a lot of research on the expression of recursive sequences,especially Lucas sequences,and got lots of results.In this thesis,the expression forms of Lucas sequences are studied through the properties of Lucas sequences themselves,combined with the decomposition of prime numbers on quadratic fields,Baker’s method,and other algebraic methods.The main results of this thesis are as follows:(1)For arbitrarily fixed Lucas sequences which can be written as the products of double factorials,it is proved by the Primitive Divisors Theorem that there is only a finite number of solutions,and then shows all the solutions to Fibonacci numbers that can be written as the products of double factorials;Secondly,it was proved that the sum or difference of two Fibonacci numbers which can be expressed as the products of double factorials are finite;Thirdly,for arbitrarily fixed Lucas sequences which can be written as sums of double factorials,it is proved by induction that there are only finite solutions;Finally,for integers N that are not Fibonacci numbers,it is proved that the number in Fibonacci numbers that can be written as a sum or difference of double factorial is finite by estimating the 5-adic of F_n-N and gives all the solutions.(2)For the reduced fraction r=p/q that is not a Fibonacci number,by estimating the 5-adic of F_n-r,it is proved that the number in the Fibonacci number that can be written as the sum or difference of repdigits and double factorial is finite,and discussed in different cases,gradually reducing the upper bound,and giving all solutions.(3)First,through Baker’s method and reduction theorem,it is proved that the Lucas number of 5 times which can be written as 2~a+3~b+5~c,where 0≤max{a,b}<≤c are finite.For the sum or difference of two Fibonacci numbers or Lucas numbers that meet the above form,the equations are established separately,and the upper bound is given by applying the Baker method,and then the reduction theorem is used to reduce to a computable upper bound,and finally,the four equations are solved computationally within the range using Mathematica software.(4)For the sum of finitely many repdigits in base b can be expressed as the power of a given prime number,it is proved by mathematical induction and algebraic linear form that there are only finitely many solutions satisfying the above conditions.