Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$

Question

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!

Answer 1

Search $n$ for which no solutions.

Q1.1.

$x^2+y^2=n^2-2\cdot4^z$

2, 881, 1762, 3524, 7048, 10467, 14096, 15713, 17841, 18511, 20511, 20934, 23623, 28192, ...

gp-code:

nxyz()=
{
 for(n=2, 10^5,
  zm= 0;
  while(2*4^zm<n^2, zm++); zm= zm-1;
  t= 0;
  for(z=1, zm,
   T= thue('x^2+1, n^2-2*4^z);
   if(#T, t= 1)
  );
  if(!t, print1(n", "))
 )
};

Q1.2.

$x^2+y^2=n^2-5\cdot4^z$

2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ...

gp-code:

  ...
  while(5*4^zm<n^2, zm++); zm= zm-1;
  ...
   T= thue('x^2+1, n^2-5*4^z);
  ...

---

Q2.1.

$x^2+2y^2=n^2-2\cdot4^z$

2, 881, 1762, 3524, 7048, 10467, 14096, 15713, 17841, 18511, 20511, 20934, 23623, 28192, 31147, ...

gp-code:

  ...
  while(2*4^zm<n^2, zm++); zm= zm-1;
  ...
   T= thue('x^2+2, n^2-2*4^z);
  ...

Q2.2.

$x^2+2y^2=n^2-3\cdot4^z$

2, 3, 5, 3215, 6430, 6917, 19187, 25717, 73413, ...

gp-code:

  ...
  while(3*4^zm<n^2, zm++); zm= zm-1;
  ...
   T= thue('x^2+2, n^2-3*4^z);
  ...