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) ∈ 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

−2 λ

x =

.

1 + λ 2

This gives us another pair of rational numbers that solve x 2 + y 2 = 1, namely

−2 λ

x =

, y = 1 − 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 = −2 λ, b = 1 − λ 2 , c = 1 + λ 2

to our original equation a 2 + b 2 = c 2.

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