Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$

Question

Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.

Answer 1

There is a parameterization for every equal sums of squares equation $$ X_1^2 + \dotsb + X_m^2 = Y_1^2 + \dotsb + Y_n^2 $$ with $n,m$ positive integers and all $X_i,Y_i$ integers. The papers by Barnett and Bradley are my first recommendations.

Answer 2

Above equation (a^2+b^2+c^2)=(d^2+e^2+f^2) has parametric solution. Refer to Tito piezas on line book "collection of algebraic Identities".
Section sum of squares. The answer is given below;

(a,b,c)=[(p+q),(r+s),(t+u)] and

(d,e,f)=[(p-q),(r-s),(t-u)]

Condition is (pq+rs+tu)=0

After parametrization of (pq+rs+tu)=0, the parametric solution in one variable is given below:

(a,b,c)=[(4k^2+12k+1),(3k-8),(7k+28)] and

(d,e,f)=[(4k^2+12k-13),(k+2),(13k+26)]

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