# Number of integer points inside an equilateral triangle with side length $n\in\mathbb{N}$

Does there exist any $$n\in\mathbb{N}$$ such that there exist at least one point inside of(not on the border) an equilateral triangle with side length $$n$$ ,which its distances to the vertices be integers?

If yes; can anyone give a formula for the number of such points for each $$n\in\mathbb{N}$$ ?

Notice: We already know by this post and for further results here ,that there are infinite number of points with rational distances inside an equilateral triangle with side length one, but what about integers?

• Well as soon as your $n$ is big enough to make the incircle have diameter $> \sqrt{2}$ you will have an integer point in your triangle... Nov 14, 2020 at 8:48
• Note that generally we can count the lattice points using Pick's area formula, e.g. as here. Nov 14, 2020 at 8:52
• @PrudiiArca so where do you choose your point location?
– MasM
Nov 14, 2020 at 9:08
• @MasM it shouldn’t matter. Any circle of diameter $>\sqrt{2}$ contains an integer point in its inside. Nov 14, 2020 at 9:10
• @BillDubuque interesting lattice points but what would be achieved here from that? the condition here seems different.
– MasM
Nov 14, 2020 at 9:10

• Great, would you explain more about how they achieved these numbers? What about a formula for the number of integer points inside a quadrilateral triangle for each $n$