Let my first blog post be about my six-year-old conjecture:
Every integer Heronian triangle is congruent to a lattice triangle.
A triangle is said to be an integer Heronian triangle, if it has integer side lengths and integer area. A triangle is said to be a lattice triangle, if all its vertices have integer coordinates.
Not much is known about this conjecture. Clearly:
- An area of any lattice triangle is an integer multiple of ½. If in addition it has integer side lengths, then its area is integer. This follows from Heron’s formula.
- Any counterexample must have a side with length greater than 1000. This was checked by enumerating all small Heronian triangles and verifying the conjecture directly.
- Noteworthy, there are integer Heronian triangles which are not congruent to a lattice triangle with one side parallel to a coordinate axis, like in the following example (side lengths 5, 29, 30):
The problem was first posed on a Polish math Usenet group pl.sci.matematyka in 2004.