Let p_0(n) denote the number of partitions counted by the paper's function p_0. Let ell>3 be a prime number with ell notequiv23 pmod24, and let delta_ell be the least positive integer satisfying 24 delta_ell equiv1 pmodell. Mod 4 congruence conjecture. For every n≥0 and every integer j with 0≤ j≤ell-2, p_0 big(2 ell^2n+2 ell j+2 delta_ell big)equiv0 pmod4. This conjecture predicts infinite arithmetic families of divisibility-by-four congruences for the partition function p_0 and motivates further investigation of its arithmetic properties.

A mod 4 congruence for partitions with parts separated by parity

From papers

Let $p_0(n)$ denote the number of partitions counted by the paper's function $p_0$ . Let $\ell>3$ be a prime number with $\ell\not\equiv23\pmod{24}$ , and let $\delta_\ell$ be the least positive integer satisfying

24\delta_\ell\equiv1\pmod{\ell}.

Mod 4 congruence conjecture. For every $n\geq0$ and every integer $j$ with $0\leq j\leq\ell-2$ ,

p_0\big(2\ell^2n+2\ell j+2\delta_\ell\big)\equiv0\pmod{4}.

This conjecture predicts infinite arithmetic families of divisibility-by-four congruences for the partition function $p_0$ and motivates further investigation of its arithmetic properties.

Progress summary

Open

The conjecture remains without a verified proof or disproof in the retrieved public literature.

The conjecture predicts that specified values of the partition function $p_0$ are divisible by four. It appears as Conjecture 5.1 in an arXiv preprint, and the scan found no proof, counterexample, claimed resolution, or verification.

Current status (as of August 2026): The statement remains a conjecture, with no retrieved public proof or disproof.

Sources

ar5iv.labs.arxiv.org

Sources & referencesView supporting material

Primary source

Shishuo Fu and Dazhao Tang, “Partitions with parts separated by parity: conjugation, congruences and the mock theta functions”, arXiv:2306.13309 (2023).

1 week ago

The conjecture fails for every admissible prime ℓ≥29, and for each such prime it fails for infinitely many n. In fact the relevant partition values are odd, so the proposed congruence fails even modulo 2.

Two results already proved in the source are decisive:

(1) Equation (4.2) states that p_ψ(m) is odd if and only if

24m−1=P^{4a+1}s²

for a prime P with P∤s.

(2) Remark 4.5 gives

p₀(2m)≡p_ψ(m) (mod 2).

Now fix any prime ℓ≥29 with ℓ≢23 (mod 24). Let δ be the least positive integer satisfying 24δ≡1 (mod ℓ), and define

r=(24δ−1)/ℓ, j=(23ℓ−r)/24.

Here 1≤r≤23. Since ℓ²≡1 (mod 24) and ℓr≡−1 (mod 24), we have r≡−ℓ (mod 24); hence j is an integer. Moreover j≥0. The same congruence shows that ℓ+r is a positive multiple of 24; because ℓ≥29, necessarily ℓ+r≥48. Therefore

j≤ℓ−2,

so j lies in precisely the range required by the conjecture.

For every n≥0 put

m=ℓ²n+ℓj+δ.

Direct substitution gives

24m−1 =24ℓ²n+24ℓj+24δ−1 =ℓ²(24n+23).

Whenever P=24n+23 is prime, we have P≠ℓ because ℓ≢23 (mod 24). Consequently

24m−1=Pℓ²

has the form in (1), with a=0 and s=ℓ. It follows that

p₀(2ℓ²n+2ℓj+2δ) ≡p_ψ(m) ≡1 (mod 2).

Dirichlet's theorem supplies infinitely many primes P≡23 (mod 24), hence infinitely many such n for every admissible ℓ≥29.

For a concrete exact instance, take

ℓ=29, δ=23, r=19, j=27, n=0.

Then the conjectured argument is

2·29·27+2·23=1612,

and the source's defining generating function

∑{v≥0}p₀(v)q^v =∑{h≥1}q^{2h²}∏_{a=1}^h(1−q^{2a−1})^{−2}

gives the exact coefficient

p₀(1612)=132350955536194689031169297511≡3 (mod 4).

Thus the published mod-4 conjecture is false both at this explicit first example and throughout the infinite all-prime family above.

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Shivam Patel · 21 hours ago