It is well known that $(x^2+y^2)(u^2+v^2)=(xu+yv)^2 + (xv-uy)^2$, thus it suffices to characterize the primes $p$ with $x^2+y^2=p$. Are there any similar composition formulae for $ax^2+by^2$ and $ax^2-by^2$?
2026-03-28 01:47:20.1774662440
Composition formula for $ax^2 \pm by^2$
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$$q(ac+nbd)^2+n(ad-qbc)^2=(a^2+nqb^2)(qc^2+nd^2)$$
We must remember that then can be reduced by a common factor.