pythagorean theorem extensionsare 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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pythagorean theorem extensionsare 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 &#039;a&#039; can be there solutions to the equation      ∑          n      =      1              N            x          i              2        =      a    2  for N=2 this is pythagorean theorem

Tanner Hamilton · Accepted Answer

Just to avoid notation bloat, let&#039;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

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

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