Work

○Open Precise open statements in the catalog. Each page lists what is known, its sources and any formal statement.

1. ○Open
   Aharoni’s rainbow cycle conjecture

   VariantCaccetta–Häggkvist cycle boundHompe & Huynh 2025

2. ○Open
   BSD rank equality holds for every E/Q

   Main statementBirch and Swinnerton-Dyer conjectureper Wiles (official description)

3. ○Open
   classical RH fails

   Possible answerRiemann hypothesisper Bombieri (official description)

4. ○Open
   classical RH holds

   Main statementRiemann hypothesisper Bombieri (official description)

5. ○Open
   Exponential Time Hypothesis

   VariantP versus NPImpagliazzo & Paturi 2001

6. ○Open
   Gaps of two infinitely often

   Main statementHow small can prime gaps be?Polymath 2014

7. ○Open
   Generalized RH for Dirichlet L-functions

   VariantRiemann hypothesisAlpöge & Furman 2026

8. ○Open
   NP ≠ coNP

   VariantP versus NPCook

9. ○Open
   NP lacks polynomial-size general circuits

   VariantP versus NPWigderson 2019

10. ○Open
    P = NP

    Main statementP versus NPCook

11. ○Open
    P ≠ NP

    Possible answerP versus NPCook

12. ○Open
    rational Hodge holds in all codimensions

    Main statementHodge conjectureper Deligne (official description)

13. ○Open
    SAT in polynomial time

    Equivalent formulationP versus NPCook

14. ○Open
    Separate conjecture: all nontrivial zeros are simple

    VariantRiemann hypothesisLamzouri 2026

15. ○Open
    Sha is finite for every E/Q

    VariantBirch and Swinnerton-Dyer conjectureper Wiles (official description)

16. ○Open
    Standard one-way functions exist

    VariantP versus NPWigderson 2019

17. ○Open
    Strong Exponential Time Hypothesis — randomized version

    VariantP versus NPCygan et al. 2014

18. ○Open
    the classical Newman constant is zero

    Equivalent formulationRiemann hypothesisRodgers & Tao 2018

19. ○Open
    the full BSD formula holds for every E/Q

    VariantBirch and Swinnerton-Dyer conjectureper Wiles (official description)

20. ○Open
    The full directed cycle bound

    Main statementCaccetta–Häggkvist cycle boundHompe & Huynh 2025

21. ○Open
    The general quadratic reset-word conjecture

    Main statementČerný synchronization conjectureZhu 2026

22. ○Open
    The one-third directed triangle question

    VariantCaccetta–Häggkvist cycle boundHladký et al. 2017

23. ○Open
    The remaining two-variable question

    VariantGaussian moments conjectureLong 2026

24. ○Open
    The sharper origin-minimum curvature conjecture

    VariantStrong log-concavity of the Chernoff densityBalabdaoui & Wellner 2014

25. ○Open
    The smooth four-dimensional Poincaré question

    VariantPoincaré conjectureManolescu 2026

26. ○Open
    the square-root prime-counting error holds

    Equivalent formulationRiemann hypothesisper Bombieri (official description)

27. ○Open
    The two-variable Jacobian question

    VariantJacobian conjectureGao 2026

28. ○Open
    Unforced global regularity in the whole space

    Main statementNavier–Stokes existence and smoothnessper Fefferman (official description)

29. ○Open
    Unforced global regularity on the three-torus

    VariantNavier–Stokes existence and smoothnessper Fefferman (official description)

30. ○Open
    VP versus VNP

    VariantP versus NPWigderson 2019