Magic Squaresproof atlas
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General proof chapter

Blue lemma: the norm in Q(√−2)

A blue four-cell support encodes a weighted equality of squares obtained by composing the norm u² + 2v².

01u=ac+2bd,v=ad−bc,w=ac−2bd,z=ad+bcu=ac+2bd,\quad v=ad-bc,\quad w=ac-2bd,\quad z=ad+bc
02u2+2v2=w2+2z2=(a2+2b2)(c2+2d2)u^2+2v^2=w^2+2z^2=(a^2+2b^2)(c^2+2d^2)
03(U,V,W,Z)=(u2,v2,w2,z2)⟹U+2V=W+2Z(U,V,W,Z)=(u^2,v^2,w^2,z^2)\Longrightarrow U+2V=W+2Z
The coefficients 1 and 2 explain why a blue support differs from an ordinary equality of two sums of squares.