n-square Identities

This page lists all the proven n-square Identities.

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Main article: Diophantus' Identity

It is known as the Diophantus' Identity or the Brahmagupta–Fibonacci identity. It states as follows:

If two positive integers are each the sum of two squares, then their product is the sum of two squares.

Consider four integers, \(a, b, c\), and \(d\).

\[\begin{align} (a^2 + b^2)(c^2 + d^2) = a^2c^2 + a^2d^2 + b^2c^2 +b^2d^2 &= a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 - 2abcd + 2abcd \\ &= (a^2c^2 \pm 2abcd + b^2d^2) + (a^2d^2 \mp 2abcd + b^2c^2) \\ &= (ac \pm bd)^2 + (ad \mp bc)^2 \end{align}\]

It is known as the Euler's four-square identity. Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

\[ (a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)= \\ (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + \\ (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 + \\ (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + \\ (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2. \]

It is known as the Degen's eight-square identity

\[(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2)(b_1^2+b_2^2+b_3^2+b_4^2+b_5^2+b_6^2+b_7^2+b_8^2)=\] \[(a_1b_1 - a_2b_2 - a_3b_3 - a_4b_4 - a_5b_5 - a_6b_6 - a_7b_7 - a_8b_8)^2+\] \[(a_1b_2 + a_2b_1 + a_3b_4 - a_4b_3 + a_5b_6 - a_6b_5 - a_7b_8 + a_8b_7)^2+\] \[(a_1b_3 - a_2b_4 + a_3b_1 + a_4b_2 + a_5b_7 + a_6b_8 - a_7b_5 - a_8b_6)^2+\] \[(a_1b_4 + a_2b_3 - a_3b_2 + a_4b_1 + a_5b_8 - a_6b_7 + a_7b_6 - a_8b_5)^2+\] \[(a_1b_5 - a_2b_6 - a_3b_7 - a_4b_8 + a_5b_1 + a_6b_2 + a_7b_3 + a_8b_4)^2+\] \[(a_1b_6 + a_2b_5 - a_3b_8 + a_4b_7 - a_5b_2 + a_6b_1 - a_7b_4 + a_8b_3)^2+\] \[(a_1b_7 + a_2b_8 + a_3b_5 - a_4b_6 - a_5b_3 + a_6b_4 + a_7b_1 - a_8b_2)^2+\] \[(a_1b_8 - a_2b_7 + a_3b_6 + a_4b_5 - a_5b_4 - a_6b_3 + a_7b_2 + a_8b_1)^2\]

It is known as the Pfister's sixteen-square identity

\[ (x_1^2+x_2^2+x_3^2+\cdots+x_{16}^2)\,(y_1^2+y_2^2+y_3^2+\cdots+y_{16}^2) = z_1^2+z_2^2+z_3^2+\cdots+z_{16}^2 \] Where \(z_i\) are \[z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 - x_5 y_5 - x_6 y_6 - x_7 y_7 - x_8 y_8 + u_1 y_9 - u_2 y_{10} - u_3 y_{11} - u_4 y_{12} - u_5 y_{13} - u_6 y_{14} - u_7 y_{15} - u_8 y_{16}\] \[z_2 = x_2 y_1 + x_1 y_2 + x_4 y_3 - x_3 y_4 + x_6 y_5 - x_5 y_6 - x_8 y_7 + x_7 y_8 + u_2 y_9 + u_1 y_{10} + u_4 y_{11} - u_3 y_{12} + u_6 y_{13} - u_5 y_{14} - u_8 y_{15} + u_7 y_{16}\] \[z_3 = x_3 y_1 - x_4 y_2 + x_1 y_3 + x_2 y_4 + x_7 y_5 + x_8 y_6 - x_5 y_7 - x_6 y_8 + u_3 y_9 - u_4 y_{10} + u_1 y_{11} + u_2 y_{12} + u_7 y_{13} + u_8 y_{14} - u_5 y_{15} - u_6 y_{16}\] \[z_4 = x_4 y_1 + x_3 y_2 - x_2 y_3 + x_1 y_4 + x_8 y_5 - x_7 y_6 + x_6 y_7 - x_5 y_8 + u_4 y_9 + u_3 y_{10} - u_2 y_{11} + u_1 y_{12} + u_8 y_{13} - u_7 y_{14} + u_6 y_{15} - u_5 y_{16}\] \[z_5 = x_5 y_1 - x_6 y_2 - x_7 y_3 - x_8 y_4 + x_1 y_5 + x_2 y_6 + x_3 y_7 + x_4 y_8 + u_5 y_9 - u_6 y_{10} - u_7 y_{11} - u_8 y_{12} + u_1 y_{13} + u_2 y_{14} + u_3 y_{15} + u_4 y_{16}\] \[z_6 = x_6 y_1 + x_5 y_2 - x_8 y_3 + x_7 y_4 - x_2 y_5 + x_1 y_6 - x_4 y_7 + x_3 y_8 + u_6 y_9 + u_5 y_{10} - u_8 y_{11} + u_7 y_{12} - u_2 y_{13} + u_1 y_{14} - u_4 y_{15} + u_3 y_{16}\] \[z_7 = x_7 y_1 + x_8 y_2 + x_5 y_3 - x_6 y_4 - x_3 y_5 + x_4 y_6 + x_1 y_7 - x_2 y_8 + u_7 y_9 + u_8 y_{10} + u_5 y_{11} - u_6 y_{12} - u_3 y_{13} + u_4 y_{14} + u_1 y_{15} - u_2 y_{16}\] \[z_8 = x_8 y_1 - x_7 y_2 + x_6 y_3 + x_5 y_4 - x_4 y_5 - x_3 y_6 + x_2 y_7 + x_1 y_8 + u_8 y_9 - u_7 y_{10} + u_6 y_{11} + u_5 y_{12} - u_4 y_{13} - u_3 y_{14} + u_2 y_{15} + u_1 y_{16}\] \[z_9 = x_9 y_1 - x_{10} y_2 - x_{11} y_3 - x_{12} y_4 - x_{13} y_5 - x_{14} y_6 - x_{15} y_7 - x_{16} y_8 + x_1 y_9 - x_2 y_{10} - x_3 y_{11} - x_4 y_{12} - x_5 y_{13} - x_6 y_{14} - x_7 y_{15} - x_8 y_{16}\] \[z_{10} = x_{10} y_1 + x_9 y_2 + x_{12} y_3 - x_{11} y_4 + x_{14} y_5 - x_{13} y_6 - x_{16} y_7 + x_{15} y_8 + x_2 y_9 + x_1 y_{10} + x_4 y_{11} - x_3 y_{12} + x_6 y_{13} - x_5 y_{14} - x_8 y_{15} + x_7 y_{16}\] \[z_{11} = x_{11} y_1 - x_{12} y_2 + x_9 y_3 + x_{10} y_4 + x_{15} y_5 + x_{16} y_6 - x_{13} y_7 - x_{14} y_8 + x_3 y_9 - x_4 y_{10} + x_1 y_{11} + x_2 y_{12} + x_7 y_{13} + x_8 y_{14} - x_5 y_{15} - x_6 y_{16}\] \[z_{12} = x_{12} y_1 + x_{11} y_2 - x_{10} y_3 + x_9 y_4 + x_{16} y_5 - x_{15} y_6 + x_{14} y_7 - x_{13} y_8 + x_4 y_9 + x_3 y_{10} - x_2 y_{11} + x_1 y_{12} + x_8 y_{13} - x_7 y_{14} + x_6 y_{15} - x_5 y_{16}\] \[z_{13} = x_{13} y_1 - x_{14} y_2 - x_{15} y_3 - x_{16} y_4 + x_9 y_5 + x_{10} y_6 + x_{11} y_7 + x_{12} y_8 + x_5 y_9 - x_6 y_{10} - x_7 y_{11} - x_8 y_{12} + x_1 y_{13} + x_2 y_{14} + x_3 y_{15} + x_4 y_{16}\] \[z_{14} = x_{14} y_1 + x_{13} y_2 - x_{16} y_3 + x_{15} y_4 - x_{10} y_5 + x_9 y_6 - x_{12} y_7 + x_{11} y_8 + x_6 y_9 + x_5 y_{10} - x_8 y_{11} + x_7 y_{12} - x_2 y_{13} + x_1 y_{14} - x_4 y_{15} + x_3 y_{16}\] \[z_{15} = x_{15} y_1 + x_{16} y_2 + x_{13} y_3 - x_{14} y_4 - x_{11} y_5 + x_{12} y_6 + x_9 y_7 - x_{10} y_8 + x_7 y_9 + x_8 y_{10} + x_5 y_{11} - x_6 y_{12} - x_3 y_{13} + x_4 y_{14} + x_1 y_{15} - x_2 y_{16}\] \[z_{16} = x_{16} y_1 - x_{15} y_2 + x_{14} y_3 + x_{13} y_4 - x_{12} y_5 - x_{11} y_6 + x_{10} y_7 + x_9 y_8 + x_8 y_9 - x_7 y_{10} + x_6 y_{11} + x_5 y_{12} - x_4 y_{13} - x_3 y_{14} + x_2 y_{15} + x_1 y_{16}\]