Direct proofs of some of Euler's results (Q2710122)

The reviewer [Invariant 1984, 3-5 (1984)] gave an elementary proof that every prime \(p\equiv 1\pmod 4\) is a sum of two squares. In particular the argument does not require one to show that \(-1\) is a quadratic residue of \(p\). An essentially similar proof was presented in a single sentence by \textit{D. Zagier} [Am. Math. Mon. 97, 144 (1990; Zbl 0735.11014)]. NEWLINENEWLINENEWLINEIn the paper under review, the author shows by similar methods that if \(p\equiv 3\pmod 8\) then \(p= a^2+ 2b^2\), and that if \(p\equiv 7\pmod {12}\) then \(p= c^2+ 3d^2\).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].