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Added (first of november 2021): A detailled proof is contained in https://hal.archives-ouvertes.fr/hal-03408135/document (also available from the arXiv in a few days).

Added (16th of october 2022): The final version, called 'A Quixotic Proof of Fermat's Two Squares Theorem for Prime Numbers', has been accepted by the American Math. Monthly. Up to formatting changes, labellings of results and minor differences it coincides with the preprint https://hal.archives-ouvertes.fr/hal-03813904 .

Added (first of november 2021): A detailled proof is contained in https://hal.archives-ouvertes.fr/hal-03408135/document (also available from the arXiv in a few days).

Sketch of proof Given a solution $(a,b,c,d)$ we consider $u=(a,c),\ v=(-d,b)$. We associate to $(a,b,c,d)$ the sublattice $\Lambda=\mathbb Zu+\mathbb Zv$ of index $p$ in $\mathbb Z^2$. Suppose now $cd>0$ and consider the eight open cones of $\mathbb R^2$ defined by the complement of the four lines defined by $xy(x^2-y^2)=0$. We colour these open cones alternatingly black and white. The four vectors $\pm u,\pm v$ are contained in different black cones (colouring the first cone above the halfline $(\mathbb R_{>0},0)$ in black).

Added (first of november 2021): A detailled proof is contained in https://hal.archives-ouvertes.fr/hal-03408135/document (also available from the arXiv in a few days).

Sketch of proof Given a solution $(a,b,c,d)$ we consider $u=(a,c),\ v=(-d,b)$. We associate to $(a,b,c,d)$ the sublattice $\Lambda=\mathbb Zu+\mathbb Zv$ of index $p$ in $\mathbb Z^2$. Suppose now $cd>0$ and consider the eight open cones of $\mathbb R^2$ defined by the complement of the four lines defined by $xy(x^2-y^2)=0$. We colour these open cones alternatingly black and white. The four vectors $\pm u,\pm v$ are contained in different black cones (colouring the first cone above the halfline $(\mathbb R_{>0},0)$ in black).