Determine the maximum number of distinct real circles tangent to each of three general real conics in the Euclidean plane. The proposed conjecture was that every such triple has at most 136 real tritangent circles, with an explicit configuration attaining 136; the 2026 preprint claims a configuration with 144.

Maximum number of circles tangent to three conics

The problem asks for the largest number of real circles tangent to three general conics. Breiding, Lindberg, Ong, and Sommer proposed the bound $136$ in 2022, but did not prove it was maximal.

Known results

Breiding, Lindberg, Ong, and Sommer (2022): $184$ complex tangent circles occur generically.
Breiding, Lindberg, Ong, and Sommer (2022): an explicit real configuration realizes $136$ circles.
Breiding, Lindberg, Ong, and Sommer (2022): computational constructions realize every even number from $0$ through $136$ .

September 2026 counterexample

A September 1, 2026 report on the preprint 144 real circles tangent to three conics states that an explicit configuration has $144$ real tritangent circles, invalidating the conjectured upper bound of $136$ . This is a claimed counterexample, not yet independently verified.

Current status (as of September 2026): The bound $136$ is claimed false because of an explicit $144$ -circle configuration, while the true maximum remains open pending independent verification.

Maximum number of circles tangent to three conics

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Additional references

Progress summary

Solutions 0