In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

EDIT: after a comment by Ofir, I'd like to ask about the case of $n$ being a prime. Already half of all primes are of the form $x^2+y^2$, so one can wonder whether finitely many forms would suffice for all primes.

In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

EDIT: after a comment by Ofir, I'd like to ask about the case of $n$ being a prime. Already half of all primes are of the form $x^2+y^2$, so one can wonder whether finitely many forms would suffice for all primes.

In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

EDIT: after a comment by Ofir, I'd like to ask about the case of $n$ being a prime. Already something like 25% of all primes are of the form $x^2+y^2$, so one can wonder whether finitely many forms would suffice for all primes.

In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

EDIT: after a comment by Ofir, I'd like to ask about the case of $n$ being a prime. Already half of all primes are of the form $x^2+y^2$, so one can wonder whether finitely many forms would suffice for all primes.

In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$ can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?

This is related to things like universal binary quadratic forms, but in my case I don't allow cross-terms $c_k xy$, and everything is nonnegative.

I imagine this must be well-known, but I can't find a reference.

EDIT: after a comment by Ofir, I'd like to ask about the case of $n$ being a prime. Already something like 25% of all primes are of the form $x^2+y^2$, so one can wonder whether finitely many forms would suffice for all primes.