Edit to open the question:
It looks like there are quadratic froms, satisfying these conditions. So, Is there any other form like quadratic form, for example, say cubic form or form of higher degree than 2 in general in the literature? What terminology used if such thing exists (so I can search )?
Previous Version:
All primes greater than $2$ can be written either in the form of $4k+1$ or $4k-1$. Fermat had successfully classified primes of the form $x^2+y^2 , x^2+ 2y^2, \text{and} \; x^2 + 3y^2$. Can all primes be written either in the form of $x^2+y^2 , x^2+ 2y^2, \text{or} \; x^2 + 3y^2$?
In general, is there a finite sets of non-trivial forms (quadratic or something else but non-linear, i.e.not like linear $an+b$ or trivial form $x^2+y$) which can be used to represent all primes?
Plz inform related topics, terminology, book, research paper in the comment, if exists.
The form $x^2 + y^2$ represents $2$ and all primes $p \equiv 1,5 \pmod 8$
The form $x^2 + 2y^2$ represents $2$ and all primes $p \equiv 1,3 \pmod 8$
The form $x^2 -2 y^2$ represents $2$ and all primes $p \equiv 1,7 \pmod 8$
Every prime is represented by at least one of the three quadratic forms.
The most suitable reference is Dickson's little 1929 book,
https://archive.org/details/in.ernet.dli.2015.466075/page/n5/mode/2up