Number Theory: An Introduction to Mathematics

576 XIII Connections with Number Theory

for some relatively prime positive integersc 1 ,d 1 .Then

(c^21 −d 12 )(c 12 +d 12 )=c 14 −d 14 =b=h^2.

But

(c 12 −d 12 ,c^21 +d^21 )= 1 ,

since(c^21 ,d 12 )=1andbis odd. Hence

c^21 −d^21 =p^2 , c^21 +d 12 =q^2 ,

for some odd positive integersp,q. Thus

a 1 =(q+p)/ 2 , b 1 =(q−p)/ 2

are positive integers and

a 12 +b^21 =(q^2 +p^2 )/ 2 =c^21 ,

2 a 1 b 1 =(q^2 −p^2 )/ 2 =d 12.

Sincec 1 ≤c 14 <c, this contradicts the minimality ofc.
It follows that the Fermat equation

x^4 +y^4 =z^4

has no solutions in nonzero integersx,y,z. For if a solution existed and if we put

u= 2 |yz|/x^2 ,v=x^2 /|yz|,w=(y^4 +z^4 )/x^2 |yz|,

we would haveu^2 +v^2 =w^2 ,uv=2.
It is easily seen that a positive integernis congruent if and only if there exists a
rational numberxsuch thatx,x+nandx−nare all rational squares. For suppose

x=r^2 , x+n=s^2 , x−n=t^2 ,

and put

u=s−t,v=s+t,w= 2 r.

Then

uv=s^2 −t^2 = 2 n

and

u^2 +v^2 = 2 (s^2 +t^2 )= 4 x=w^2.