Let $a$ be a prime of the form $20 n + 3$ or $20 n + 7$ Let $b$ be a prime of the form $20 m + 3$ or $20 m + 7$.
Then the semiprime $ S = ab $ is always of the form
$ S = x^2 + 5 y^2 $.
For some integers $x,y$.
How to prove that ?
How does elementary number theory deal with this and how would analytic number theory deal with it ?
Does every proof use modular arithmetics ?
I wonder about analogue results too. Such as a cubic case or a triple product.
Both primes are represented by $$ 2 u^2 + 2 uv + 3 v^2. $$
Note: for a discriminant $\Delta,$ and a prime $p$ such that, first, $\Delta \neq 0 \pmod p,$ and then Legendre $$ (\Delta | p) = 1, $$ a simple construction shows that there is a form of discriminant $\Delta$ that represents $p.$ Furthermore, the form can then be reduced, an easy thing for positive forms. Put together, for discriminant $-20,$ every prime with $(-20|p) = 1$ is represented by either $x^2 + 5 y^2$ or $2x^2 + 2xy+3y^2.$ There is just one form per genus: the primes $1,9 \pmod{20}$ are represented by $x^2 + 5 y^2,$ the primes $3,7 \pmod{20}$ are represented by $2x^2 +2xy+ 3 y^2.$
The representation of the product by $x^2 + 5 y^2$ is just Gauss composition.
in Cox, Primes of the form $x^2 + n y^2,$ page 37 in the first edition, he gives a formula of Lagrange,
$$ (2x^2 + 2 xy+ 3 y^2)(2z^2 + 2zw + 3 w^2) = (2xz+xw+yz+3yw)^2 +5(xw-yz)^2 $$