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

Question

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... — Jonas Linssen, Nov 14 '20 at 08:48
Note that generally we can count the lattice points using Pick's area formula, e.g. as here. — Bill Dubuque, Nov 14 '20 at 08:52
@MasM it shouldn’t matter. Any circle of diameter $>\sqrt{2}$ contains an integer point in its inside. — Jonas Linssen, Nov 14 '20 at 09:10
@BillDubuque interesting lattice points but what would be achieved here from that? the condition here seems different. — MasM, Nov 14 '20 at 09:10
Multiply rationals by the common denominator, and you'll have integers. It is as simple as that. — Ivan Neretin, Nov 14 '20 at 09:44
@IvanNeretin Sure this is what we have from that post, what is different here is that the formula for $n$ — MasM, Nov 14 '20 at 10:15

score 2 · Accepted Answer · answered Feb 03 '21 at 21:45

2

Yes, infinitely many such triangles exist. Possible n are 112, 147, 185, 224, 273, 283, 294, 331...... See https://oeis.org/A061281

answered Feb 03 '21 at 21:45

Herbert Kociemba

241

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 $n$ – MasM Feb 07 '21 at 21:33
I only can tell a you a recipe how I could reproduce these numbers. Define the coordinates P1, P2, P3 of the vertices of an equilateral triangle with edge length n in a cartesian coordinate system and the coordinate C of all triangles with integer edge lengths a<=b<n which share the edge P1P2 with the equilateral triangle and with the 3rd vertex C inside the equilateral triangle. Check if CP3 is an integer for any of these possible inner points C. – Herbert Kociemba Feb 08 '21 at 22:29
I don't think this is the way computers may calculate the solutions ,I think there would be a subtle way to come up with numerical calculations of the tremendous diophantine equation behind it. – MasM Feb 12 '21 at 08:39
@MasM Which tremendous diophantine equation? And it is exactly the way how computers can compute the solutions for n in the order about 10^3 to 10^4. I am quite sure that no explicit formula exists. – Herbert Kociemba Feb 12 '21 at 20:18

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

1 Answers1