In March 2018, I formulated the following somewhat curious question.
Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ is a sum of two squares?
See https://oeis.org/A300510 for the history of this question. In June 2019, via a computer G. Resta found that the question has a positive answer for all $n=2,\ldots,6\times10^9$.
In March 2018, I also formulated another similar question.
Question 2. Whether for any integer $n>1$ we can write $n^2$ as $x^2+2y^2+2\times4^z$ or $x^2+2y^2+3\times4^z$ with $x,y,z$ nonnegative integers?
My computation indicates that Question 2 has a positive answer for all $n=2,\ldots,5\times10^7$. See http://oeis.org/A301452.
Can one find a counterexample to Question 1 or Question 2?
Any comments are welcome!