Number Theory: An Introduction to Mathematics

562 XIII Connections with Number Theory

To prove (∗) we may evidently assume thatP 1 =(x 1 ,y 1 )andP 2 =(x 2 ,y 2 )are
both finite. Moreover, by Proposition 8, we may assume thatP 1 =P 2. Then, by the
formulas of§3,

P 1 +P 2 =(x 3 ,y 3 ), P 1 −P 2 =(x 4 ,y 4 ),

where

x 3 =(y 2 −y 1 )^2 /(x 2 −x 1 )^2 −(x 1 +x 2 ),

x 4 =(y 2 +y 1 )^2 /(x 2 −x 1 )^2 −(x 1 +x 2 ).

Hence

x 3 +x 4 =2[y^22 +y^21 −(x 2 −x 1 )(x^22 −x^21 )]/(x 2 −x 1 )^2

and

x 3 x 4 =(y 22 −y 12 )^2 /(x 2 −x 1 )^4 − 2 (x 1 +x 2 )(y 12 +y 22 )/(x 2 −x 1 )^2 +(x 1 +x 2 )^2.

Sincey^2 j=x^3 j+axj+b(j= 1 , 2 ), these relations simplify to

x 3 +x 4 =2[x 1 x 2 (x 1 +x 2 )+a(x 1 +x 2 )+ 2 b]/(x 2 −x 1 )^2

and

x 3 x 4 =N/(x 2 −x 1 )^2 ,

where

N=(x 22 +x 1 x 2 +x 12 +a)^2 − 2 (x 1 +x 2 )^2 (x 22 −x 1 x 2 +x 12 +a)

− 4 b(x 1 +x 2 )+(x^22 −x 12 )^2

=(x 1 x 2 −a)^2 − 4 b(x 1 +x 2 ).

Putxj=pj/qj,where(pj,qj)= 1 ( 1 ≤j≤ 4 ).Thenx 3 ,x 4 are the roots of the
quadratic polynomial

AX^2 +BX+C

with integer coefficients

A=(p 2 q 1 −p 1 q 2 )^2 ,

B=(p 1 p 2 +aq 1 q 2 )(p 1 q 2 +p 2 q 1 )+ 2 bq^21 q 22 ,

C=(p 1 p 2 −aq 1 q 2 )^2 − 4 bq 1 q 2 (p 1 q 2 +p 2 q 1 ).

Consequently

Ap 3 p 4 =Cq 3 q 4 ,

A(p 3 q 4 +p 4 q 3 )=Bq 3 q 4.