For any prime $p>5$, one of the numbers $$1^2+1=2,\ \ 2^2+1=5,\ \ 3^2+1=10=2\times5$$ is a quadratic residue modulo $p$. In 2014 I conjectured that each prime $p$ has a primitive root $g<p$ of the form $k^2+1\ (k\in\mathbb Z)$ (cf. http://oeis.org/A239957); this is still open.

By a result of Fermat, the equation $x^4+y^4=z^2$ has no positive integer solution.

In view of the above, here I ask the following question.

**Question 1.** Whether for each prime $p>541$ there is a number of the form $x^4+y^4$ (with $x,y\in\mathbb Z)$ which is not only smaller than $p$ but also a quadratic residue modulo $p$?

Actually, I even conjecture that for any prime $p>541$ with $p\not=941$ there is a prime $q<p$ of the form $x^4+y^4\ (x,y\in\mathbb Z)$ with $\left(\frac q p\right)=1$, where $(-)$ is the Legendre symbol. Of course, it is not yet proven that there are infinitely many primes of the form $x^4+y^4$.

The following question is similar to Question 1.

**Question 2.** Whether for each odd prime $p\not\in\{7,17,47,103\}$ there is a number $q<p$ of the form $x^4+y^4\ (x,y\in\mathbb Z)$ with $\left(\frac qp\right)=-1$?

In 2001 Heath-Brown [Acta Math. 186 (2001), 1-84] proved that there are infinitely many primes of the form $x^3+2y^3$ with $x,y\in\mathbb N=\{0,1,2,\ldots\})$. Motivated by this, here I pose the following question.

**Question 3.** Whether for each odd prime $p$ there is a prime $q<p$ with $\left(\frac qp\right)=-1$ such that $q=x^3+2y^3$ for some $x,y\in\mathbb N$ with $y+1$ prime?

I have checked Question 3 for all odd primes $p<2\times10^9$; see http://oeis.org/A344173 for related data. For example, the prime $q=3^3+2(3-1)^3=43$ is a quadratic nonresidue modulo the prime $p=457$.

Your comments are welcome!