Thomas A Garrity All The Math You Missed But Need To Know For Graduate School Cambridge University Press 2022 by Desconocido
Author:Desconocido [Desconocido]
Language: spa
Format: epub, azw3
Published: 2021-06-17T17:43:33+00:00
a 2
2
+ b
= 1.
c
c
This means that finding any integer solutions to a 2 + b 2 = c 2 corresponds to finding a pair of rational numbers x and y that solve
x 2 + y 2 = 1 ,
and thus to a rational point on the unit circle.
186
Elementary Number Theory
We want to show that there exist infinitely many solutions in Q2. We do know that x = 0, y = 1 is a solution. Consider the line
Lλ = { (x,y) â R2 : y = λx + 1}
with slope λ.
This line must intersect the circle in two points, one of which is ( 0 , 1 ). To find the other point of intersection, we need to solve
(x 2 + y 2 = 1 ) ⩠(y = λx + 1 ) which means we must solve
x 2 + (λx + 1 ) 2 = 1 ,
which is the quadratic equation
( 1 + λ 2 )x 2 + 2 λx = 0.
Thus the two roots are x = 0, which we already knew, and
â2 λ
x =
.
1 + λ 2
This gives us another pair of rational numbers that solve x 2 + y 2 = 1, namely
â2 λ
x =
, y = 1 â λ 2
1 + λ 2
1 + λ 2
provided that the slope λ is a rational number. Thus each rational slope λ gives us a new rational solution to x 2 + y 2 = 1, giving us the integer solution a = â2 λ, b = 1 â λ 2 , c = 1 + λ 2
to our original equation a 2 + b 2 = c 2.
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