6 Some Applications 577Conversely, ifu,v,ware rational numbers such thatuv= 2 nandu^2 +v^2 =w^2 ,then
(u+v)^2 =w^2 + 4 n,(u−v)^2 =w^2 − 4 n.Thus, if we putx=(w/ 2 )^2 ,thenx,x+nandx−nare all rational squares.
It may be noted that ifxis a rational number such thatx,x+nandx−nare all
rational squares, thenx=−n, 0 ,n,sincen>0 and 2 is not a rational square.
The problem of determining which positive integers are congruent was considered
by Arab mathematicians of the 10th century AD, and later by Fibonacci (1225) in his
Liber Quadratorum. The connection with elliptic curves will now be revealed:
Proposition 13A positive integer n is congruent if and only if the cubic curve Cn
defined by the polynomial
Y^2 −(X^3 −n^2 X)has a rational point P=(x,y)with y= 0.
Proof Suppose first thatnis congruent. Then there exists a rational numberxsuch
thatx,x+nandx−nare all rational squares. Hence their product
x^3 −n^2 x=x(x−n)(x+n)is also a rational square. Sincex=−n, 0 ,n, it follows thatx^3 −n^2 x=y^2 ,wherey
is a nonzero rational number.
Suppose now thatP=(x,y)is any rational point of the curveCnwithy=0. If
we put
u=|(x^2 −n^2 )/y|,v=| 2 nx/y|,w=|(x^2 +n^2 )/y|,thenu,v,ware positive rational numbers such that
u^2 +v^2 =w^2 , uv= 2 n. It is readily verified thatλ= 1 /2 in the Riemann normal form forCn.
We now show that, for every positive integern, the torsion group ofCnhas order 4,
consisting of the identity elementO, and the three elements( 0 , 0 ),(n, 0 ),(−n, 0 )of
order 2. Assume on the contrary that for some positive integernthe curveCnhas a
rational pointP=(x,y)of finite order withy=0andtakento be the least positive
integer with this property. Then 2P=(x′,y′)is also a rational point ofCnof finite
order. The formula for the other point of intersection withCnof the tangent toCnat
Pshows that
x′=[(x^2 +n^2 )/ 2 y]^2.It follows that
x′+n=[(x^2 −n^2 + 2 nx)/ 2 y]^2 ,
x′−n=[(x^2 −n^2 − 2 nx)/ 2 y]^2.