In this paper,we study Diuphantion equation 2 a+2 b+2 c= x 2,and the sufficient and necessary condition for 2 a+2 b+2 c as square number is given.

In this paper, the author studies the condition for square numbers in a second order recursion sequence:L_0=0,L_1=1,L_(n+2)=2kL_(n+1)-L_n (n≥0)V_0=2,V_1=2k,V_(n+2)=2kV_(n+1)-V_n (n≥0)And some results are reached by solving several Diophantine equafions.

All positive integers which make numbers with the form 1+25n(n+1)/2 to be square numbers were given by using the basic knowledge of the Pell equation in this paper.

This article the object studied is not only a triangular number,but also the square numbers "the triangle square numbers".

The problem of transforming k-angular numbers into square numbers is discussed.

Dual Consecutive Number is the Necessary and Sufficient Condition of a Square Integer;

Recurring decimal depicts positive integer solution of x and y in a indefinite equation of x2=(10k+1)y which first that square integer is a necessary and sufficient condition of consecutive number,and then shows all the dual consecutive number of a square integer,also denied the suspect of passage [1].

In this paper,all positive integers n which makes the form 1+9n(n+1)/2 to be a square was given.

An infinite product and its square are expanded to infinite series by residue theorem.

On certain divisor sum over square-full integers;

Some asymptotic fomulas on square complements;

On the mean value of divisor function for square complements;

The hybrid mean value of square complement and the sum of divisors function