I cannot do this!

Number Theory Level 4

$x$ and $y$ are positive integers which satisfy:

$4x^2-y^2=315$

Suppose all of its solution can be stated in the form $(x,y) = (x_i, y_i)$ .

What is the value of $\sum y_i - \sum x_i$ ?

1 solution

Shubham Bhuyan
Dec 17, 2014

Given equations (2x)² -y² = 315

Which is (2x+y)(2x-y)=315

Factorise 315.... The factors are 3,3,5,7

Take all possible combination I.e 1*315
3*105
5*63
7*45
35*9
& 21*15

1*315=(158-157)(158+157)
3*105=(54-51)(54+51)
63*5= (34+29)(34-29)
45*7=(26+19)(26-19)
35*9=(22+13)(22-13)
21*15=(18+3)(18-3)

Now that all the equations are in (a+b)(a-b) form We can say 2x= 158 & y=157 so x=79.

we find all possible values of x as (79, 27,17,13,11,9) corresponding y values are 157,51,29,19,13,3

Sigma(y values)-sigma(x values) = 272-156= 116