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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?

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

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

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  • $\begingroup$ 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 $n$ $\endgroup$
    – MasM
    Commented Feb 7, 2021 at 21:33
  • $\begingroup$ 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. $\endgroup$ Commented Feb 8, 2021 at 22:29
  • $\begingroup$ 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. $\endgroup$
    – MasM
    Commented Feb 12, 2021 at 8:39
  • $\begingroup$ @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. $\endgroup$ Commented Feb 12, 2021 at 20:18

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