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Pfister's sixteen-square identity

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In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form

( x 1 2 + x 2 2 + x 3 2 + ⋯ + x 16 2 ) ( y 1 2 + y 2 2 + y 3 2 + ⋯ + y 16 2 ) = z 1 2 + z 2 2 + z 3 2 + ⋯ + z 16 2 {\displaystyle \left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{16}^{2}\right)\left(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{16}^{2}\right)=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{16}^{2}}

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is

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 {\displaystyle {\begin{aligned}&\scriptstyle {z_{1}={\color {blue}{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}}\\&\scriptstyle {z_{2}={\color {blue}{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}}\\&\scriptstyle {z_{3}={\color {blue}{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}}\\&\scriptstyle {z_{4}={\color {blue}{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}}\\&\scriptstyle {z_{5}={\color {blue}{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}}\\&\scriptstyle {z_{6}={\color {blue}{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}}\\&\scriptstyle {z_{7}={\color {blue}{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}}\\&\scriptstyle {z_{8}={\color {blue}{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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\\&\scriptstyle {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}}\end{aligned}}}

If all x i {\displaystyle x_{i}} and y i {\displaystyle y_{i}} with i > 8 {\displaystyle i>8} are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The u i {\displaystyle u_{i}} are

u 1 = ( a x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 9 − 2 x 1 ( b x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 2 = ( x 1 2 + a x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 10 − 2 x 2 ( x 1 x 9 + b x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 3 = ( x 1 2 + x 2 2 + a x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 11 − 2 x 3 ( x 1 x 9 + x 2 x 10 + b x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 4 = ( x 1 2 + x 2 2 + x 3 2 + a x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 12 − 2 x 4 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + b x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 5 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + a x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 13 − 2 x 5 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + b x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 6 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + a x 6 2 + x 7 2 + x 8 2 ) x 14 − 2 x 6 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + b x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u 7 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + a x 7 2 + x 8 2 ) x 15 − 2 x 7 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + b x 7 x 15 + x 8 x 16 ) c u 8 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + a x 8 2 ) x 16 − 2 x 8 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + b x 8 x 16 ) c {\displaystyle {\begin{aligned}&u_{1}={\tfrac {\left(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{9}-2x_{1}\left(bx_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{2}={\tfrac {\left(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{10}-2x_{2}\left(x_{1}x_{9}+bx_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{3}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+ax_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{11}-2x_{3}\left(x_{1}x_{9}+x_{2}x_{10}+bx_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{4}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+ax_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{12}-2x_{4}\left(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+bx_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{5}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+ax_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{13}-2x_{5}\left(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+bx_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{6}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+ax_{6}^{2}+x_{7}^{2}+x_{8}^{2}\right)x_{14}-2x_{6}\left(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+bx_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{7}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+ax_{7}^{2}+x_{8}^{2}\right)x_{15}-2x_{7}\left(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+bx_{7}x_{15}+x_{8}x_{16}\right)}{c}}\\&u_{8}={\tfrac {\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+ax_{8}^{2}\right)x_{16}-2x_{8}\left(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+bx_{8}x_{16}\right)}{c}}\end{aligned}}}

and,

a = − 1 , b = 0 , c = x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 . {\displaystyle a=-1,\;\;b=0,\;\;c=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\,.}

The identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. Incidentally, the u i {\displaystyle u_{i}} also obey,

u 1 2 + u 2 2 + u 3 2 + u 4 2 + u 5 2 + u 6 2 + u 7 2 + u 8 2 = x 9 2 + x 10 2 + x 11 2 + x 12 2 + x 13 2 + x 14 2 + x 15 2 + x 16 2 {\displaystyle u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2}+u_{5}^{2}+u_{6}^{2}+u_{7}^{2}+u_{8}^{2}=x_{9}^{2}+x_{10}^{2}+x_{11}^{2}+x_{12}^{2}+x_{13}^{2}+x_{14}^{2}+x_{15}^{2}+x_{16}^{2}}

No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form

( x 1 2 + x 2 2 + x 3 2 + ⋯ + x n 2 ) ( y 1 2 + y 2 2 + y 3 2 + ⋯ + y n 2 ) = z 1 2 + z 2 2 + z 3 2 + ⋯ + z n 2 {\displaystyle \left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+\cdots +y_{n}^{2}\right)=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots +z_{n}^{2}}

with the z i {\displaystyle z_{i}} bilinear functions of the x i {\displaystyle x_{i}} and y i {\displaystyle y_{i}} is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the z i {\displaystyle z_{i}} are rational functions of one set of variables, hence has a denominator, then it is possible for all n = 2 m {\displaystyle n=2^{m}} . There are also non-bilinear versions of Euler's four-square and Degen's eight-square identities.

See also

References

  1. H. Zassenhaus and W. Eichhorn, "Herleitung von Acht- und Sechzehn-Quadrate-Identitäten mit Hilfe von Eigenschaften der verallgemeinerten Quaternionen und der Cayley-Dicksonchen Zahlen," Arch. Math. 17 (1966), 492-496
  2. A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Körper," J. London Math. Soc. 40 (1965), 159-165
  3. Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf

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