With a recursive sequence,quadratic remainder and congruence,the diophantine equation x2-3y4=97 is proved that it has only positive integral solutions(x,y)=(10,1).

Let p be a prime number,using Fermat Infinite method of descent,to study the positive integral solution of the equations x~4±3px~2y~2+3p~2y~4=z~(2) and x~4±6px~2y~2-3p~2y~4=z~(2).

By using computer language, we get all the positive integral solution of x2±xy+y2 = P and x2±xy+y2 = 3p within the scope of arbitrarily.

An equation involving the pseudo Smarandache function and its positive integer solutions;

On the necessary condition of one class of hyperelliptic equations having the positive integer solutions;

A function equation related to the Smarandache function and its positive integer solutions;

The equation xn+yn=zn(n>2)(The letters are all positive integer except especially mentioned) was proven by quadratic equation.

Let n be positive integer,f(n) here refers to the mininum mumber of 1 when n can be expressed by 1,+,× and().

Based on the literature([1]),we apply 1+12~2+13~2+…+=π~26 to study the estimation of any subdivide numbers of a positive integer where N and the subdivide numbers allow to be disorder and repeated.