TITLE: How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?
QUESTION [3 upvotes]: A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?
I thought there were two points, but that isn't right. What am I missing?

REPLY [3 votes]: Remark(I): The two equations intersects at $(\pm 2, 2)$.  
Remark(II): The points $(\pm 2, 2); \  (\pm 1, 5); \ (0, 6)$ lies 
on the boundary of the parabola $y=-x^2+6$.  
Remark(III): The points $(\pm 2, 2); \  (\pm 1, 1); \ (0, 0)$ lies 
on the boundary of $y= | x |$.  

Remark(IV): Consider the line $y=1$; 
we know that the point $(1,5)$ 
lies on the boundary of the parabola $y=-x^2+6$; 
also 
we know that the point $(1,1)$ 
lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; 
which lies on the line $y=1$; are in the interior, 
i.e. all the points $(1, 2); \ (1, 3); \ (1, 4)$ are in the interior. 
Remark(V): Consider the line $y=-1$; 
we know that the point $(-1,5)$ 
lies on the boundary of the parabola $y=-x^2+6$; 
also 
we know that the point $(-1,1)$ 
lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; 
which lies on the line $y=-1$; are in the interior, 
i.e. all the points $(-1, 2); \ (-1, 3); \ (-1, 4)$ are in the interior. 

Remark(VI): Consider the line $y=0$; 
we know that the point $(0,6)$ 
lies on the boundary of the parabola $y=-x^2+6$; 
also 
we know that the point $(0,0)$ 
lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; 
which lies on the line $y=0$; are in the interior, 
i.e. all the points 
$(0, 1); \ (0, 2); \ (0, 3) \ (0, 4); \ (0, 5); $ are in the interior. 




The points on the boundary of are as follows:
$$(\pm 2, 2); \  (\pm 1, 5); \ (0, 6); 
\\ 
(\pm 1, 1); \ (0, 0).  $$ 

The interior points are as follows:
$$(\pm 1, 2); \ (\pm 1, 3); \ (\pm 1, 4); 
\\ 
(0, 1); \ (0, 2); \ (0, 3); \ (0, 4); \ (0,5).  $$ 

So there are $19$ such points.
{"set_name": "stack_exchange", "score": 3, "question_id": 2406780}