pythagorean theorem extensions are there for a given integer N solutions to the equations <mund

kramberol 2022-07-11 Answered
pythagorean theorem extensions
are there for a given integer N solutions to the equations
n = 1 N x i 2 = z 2
for integers x i and zan easier equation given an integer number 'a' can be there solutions to the equation
n = 1 N x i 2 = a 2
for N=2 this is pythagorean theorem
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Answers (1)

Tanner Hamilton
Answered 2022-07-12 Author has 12 answers
Just to avoid notation bloat, let's let N=4 and let the implied generalization take care of the rest of the question. The question can be restated as asking whether there are rational points on the 4-sphere S 4 : x 1 2 + x 2 2 + x 3 2 + x 4 2 = 1. We know that (1,0,0,0) is a (trivial) rational point on S4. From that point, we pick a rational direction, ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) where ξ 1 , ξ 2 , ξ 3 , ξ 4 are rational numbers, and see if the line ( 1 , 0 , 0 , 0 ) + t ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) intersects S4 at another rational point.
( 1 + t ξ 1 ) 2 + t 2 ξ 2 2 + t 2 ξ 3 2 + t 2 ξ 4 1 = 1 2 t ξ 1 + t 2 ( ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 ) = 0 t = 2 ξ 1 ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2
So, for any four rational numbers ξ 1 , ξ 2 , ξ 3 , ξ 4
( ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 , 2 ξ 1 ξ 2 ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 , 2 ξ 1 ξ 3 ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 , 2 ξ 1 ξ 4 ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 , )
is a rational point on S4
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