Completeness conjecture for triangles satisfying Pick's formula almost correctly

About 20 years old · traced to 2006

Let $p,q \in \mathbb{Z}$ be coprime integers with $1<q<p^2$ , with $p$ odd and $q$ even. Let $\Delta(p,q)$ denote the triangle defined in the paper, and suppose that Pick's formula is almost correct for every dilate $r\Delta(p,q)$ with $r\in\mathbb{N}$ . Completeness conjecture. The pair $(p,q)$ belongs to one of the four Casson–Gordon families stated above. This conjecture asserts that the known Casson–Gordon families exhaust all pairs for which Pick's formula is almost correct for every dilate; the paper presents empirical evidence but does not establish completeness.

References

Primary source

Michael Eisermann and Christoph Lamm, “For which triangles is Pick's formula almost correct?”, arXiv:math/0602393 (2009).

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Open

The conjecture remains open: computations support the proposed list of four families, but no proof or counterexample has been found.

Eisermann and Lamm’s 2006 paper formulates the conjecture that the four known Casson–Gordon families exhaust all pairs whose triangles satisfy the stated Pick-formula condition for every dilate.

Known results

Casson and Gordon verified the conjecture computationally for $p \le 105$ .
Eisermann and Lamm extended the computation to $p<5000$ .
The paper states that all four Casson–Gordon families satisfy the condition for every dilate.
Lisca’s classification of two-bridge ribbon knots solves a related topological problem, not this lattice-point conjecture.

Current status (as of September 2026): The conjecture remains open; finite computations support completeness, but no proof or counterexample is recorded.

Sources

No solutions have been posted yet.