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Representations of primes by simple quadratic forms, such as $\pm a^2\pmqb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gaussand many others. We are interested in a comprehensive list of such results, for$q\le 20$. Some of the results can be established with elementary methods andwe put them at work on some instances. We are introducing new relationshipsbetween various representations.

Primes of the form \pm a^2\pm qb^2