Parities of partition numbers

The question

parkin_shanks. The natural density of n where the partition number p(n) is even (resp. odd) exists and equals 1/2.

ono.polynomial. Ono's conjecture, as stated in Section 1 of Ono–Swaminathan: if g is a nonconstant integer-valued polynomial that is eventually positive, then p(g(n)) is even for infinitely many n and odd for infinitely many n.

There are several conjectures on parities of partition numbers. The most famous conjecture is the Parkin-Shanks conjecture, which states that the natural density of n where the partition number p(n) is even (resp. odd) exists and equals 1/2.

There are several related (weaker) results toward the conjecture. Kolberg showed that p(n) takes each parity infinitely often. Subbarao conjectured that every arithmetic progression r (mod t) contains infinitely many n with p(n) even and infinitely many n with p(n) odd; the even case was proved by Ono and the odd case by Radu. Quantitative lower bounds for the number of n ≤ X with p(n) even (resp. odd) have also been studied. The best known bounds are ≫ √X log log X for even values (Bellaïche–Nicolas) and ≫ √X for odd values (Zheng, in an appendix to Ono–Swaminathan). Ono–Swaminathan also proved that both parities occur infinitely often along the quadratic progressions (D m² + 1) / 24, and Ono conjectured the same for every nonconstant integer-valued polynomial that is eventually positive.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.PartitionParity (2 statements).

theorem parkin_shanks :
    {n : ℕ | Even (partitionNumber n)}.HasDensity (1 / 2) ∧
    {n : ℕ | Odd (partitionNumber n)}.HasDensity (1 / 2)

theorem ono.polynomial (g : ℚ[X]) (hg : 0 < g.natDegree)
    (hint : ∀ n : ℕ, ∃ k : ℤ, g.eval (n : ℚ) = k)
    (hpos : ∀ᶠ n : ℕ in atTop, 0 < g.eval (n : ℚ)) :
    {n : ℕ | Even (partitionNumber ⌊g.eval (n : ℚ)⌋₊)}.Infinite ∧
    {n : ℕ | Odd (partitionNumber ⌊g.eval (n : ℚ)⌋₊)}.Infinite

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• On the distribution of parity in the partition function, T. R. Parkin, D. Shanks, Math. Comp. 21 (1967), 466–480.
• Note on the parity of the partition function, O. Kolberg, Math. Scand. 7 (1959), 377–378.
• Some remarks on the partition function, M. V. Subbarao, Amer. Math. Monthly 73 (1966), 851–854.
• Parity of the partition function in arithmetic progressions, K. Ono, J. Reine Angew. Math. 472 (1996), 1–16.
• Distribution of parity of the partition function in arithmetic progressions80014-7), S. Ahlgren, Indag. Math. (N.S.) 10 (1999), 173–181.
• Parity of the partition function, K. Ono, Adv. Math. 225 (2010), 349–366.
• A proof of Subbarao's conjecture, C.-S. Radu, J. Reine Angew. Math. 672 (2012), 161–175.
• Parité des coefficients de formes modulaires, J. Bellaïche, J.-L. Nicolas, Ramanujan J. 40 (2016), 1–44.
• Nonzero coefficients of half-integral weight modular forms mod ℓ, J. Bellaïche, B. Green, K. Soundararajan, Res. Math. Sci. 5 (2018), Paper No. 6.
• Parity of the partition function in quadratic progressions (with an appendix by Q.-Y. Zheng), K. Ono, A. Swaminathan, arXiv:2509.09553.
• [Cox] D. A. Cox, Primes of the Form $x^2+n{y}^2$, 2nd ed., Wiley, 2013.

Source and licence

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