---
title: Swee Hong Chan's Conjecture
url: https://www.emergentmind.com/topics/swee-hong-chan-s-conjecture
type: topic
---

# Swee Hong Chan's Conjecture

“Swee Hong Chan’s conjecture” denotes, in the arXiv literature provided here, a family of conjectural statements due to S. H. Chan rather than a single universally fixed proposition. The two most direct and internally coherent usages are number-theoretic and combinatorial. In the first, Chan conjectured explicit congruences for coefficients attached to Ramanujan’s sixth order mock theta function and to related Appell–Lerch sums; these conjectures were proved by Baruah and Begum, who also obtained stronger congruences. In the second, Chan conjectured a bijection between necklaces of length \(n\) with at most \(q\) colors and periodic functions \(f:\mathbb Z_n\to\{0,1,\dots,q-1\}\) whose weighted sum is divisible by \(n\), under the coprimality hypothesis \(\gcd(n,q)=1\); a 2025 paper presents a proof of this bijection [1807.03957], [2507.19940].

## 1. Principal formulations

Within the supplied literature, the most direct formulations associated with S. H. Chan are as follows.

| Context | Conjectural content | Resolution |
|---|---|---|
| Mock theta/Appell–Lerch sums | Congruences for \(a(n)\), \(a_{1,10}(n)\), and \(a_{3,10}(n)\) in specified arithmetic progressions | Proved; stronger mod \(125\) congruences also obtained |
| Necklaces and restricted multisets | A bijection for \(\gcd(n,q)=1\) between necklaces and periodic functions with weighted sum divisible by \(n\) | Proved |

In the mock-theta setting, the basic coefficient sequence is defined by
\[
\sum_{n=1}^\infty a(n)q^n:=\phi(q),
\]
where \(\phi(q)\) is Ramanujan’s function, identified in the paper as one of the sixth order mock theta functions. Chan had previously proved
\[
a(10n+9)\equiv 0 \pmod 5,
\]
and conjectured stronger congruences modulo \(25\). He also conjectured congruences for coefficients of generalized Appell–Lerch sums \(a_{j,p}(n)\) [1807.03957].

In the necklace setting, the conjecture is stated as a bijection between necklaces of length \(n\) with at most \(q\) colors and periodic functions \(f:\mathbb Z_n\to\{0,1,\dots,q-1\}\) whose weighted sum is divisible by \(n\), with the essential hypothesis
\[
\gcd(n,q)=1.
\]
The 2025 paper is presented as a proof of this conjecture [2507.19940].

## 2. Congruence conjectures for sixth order mock theta coefficients

The number-theoretic form of Chan’s conjecture arises from Ramanujan’s sixth order mock theta function \(\phi(q)\) and related Appell–Lerch sums. The standard \(q\)-Pochhammer notation is used:
\[
(a;q)_0:=1,\qquad (a;q)_n:=\prod_{k=0}^{n-1}(1-aq^k)\quad (n\ge 1),
\]
and
\[
(a_1,a_2,\dots,a_r;q)_n:=(a_1;q)_n(a_2;q)_n\cdots(a_r;q)_n.
\]
The paper also adopts the shorthand \(E_j:=(q^j;q^j)_\infty\) [1807.03957].

Chan’s conjectured congruences for \(a(n)\) are
\[
a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv 0 \pmod{25}.
\]
These strengthen the previously established congruence modulo \(5\),
\[
a(10n+9)\equiv 0 \pmod 5.
\]
The same paper also treats generalized Appell–Lerch sums \(a_{j,p}(n)\), noting in particular that
\[
2a(n)=a_{1,2}(n).
\]
For these coefficients, Chan conjectured
\[
a_{1,10}(10n+5)\equiv 0 \pmod 5,\qquad a_{3,10}(10n+5)\equiv 0 \pmod 5.
\]

These statements place Chan’s conjecture in the arithmetic theory of mock theta functions, where the central phenomenon is congruence behavior in specific residue classes. The conjectures are explicit, progression-by-progression divisibility assertions rather than asymptotic or structural hypotheses.

## 3. Proof methods and arithmetic consequences

Baruah and Begum prove Chan’s congruences by deriving an exact generating function for the progression \(a(10n+9)\) in terms of \(q\)-products, then reducing it modulo \(5\), applying
\[
E_5\equiv E_1^5 \pmod 5,
\]
and using Jacobi’s identity
\[
E_1=\sum_{k=0}^\infty (-1)^k(2k+1)q^{k(k+1)/2}.
\]
From coefficient extraction in residue classes \(q^{5n+r}\), they obtain
\[
a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv 0 \pmod{25}.
\]
They then push the same generating-function method further to prove the new congruences
\[
a(1250n+250r+219)\equiv 0 \pmod{125},\qquad r=1,3,4.
\]
These modulo \(125\) congruences are explicitly described as new [1807.03957].

For \(a_{1,10}(n)\) and \(a_{3,10}(n)\), the proof begins from identities obtained from Qu–Wang–Yao’s general theorem and proceeds via Ramanujan’s general theta function
\[
f(a,b):=\sum_{k=-\infty}^\infty a^{k(k+1)/2}b^{k(k-1)/2},\qquad |ab|<1,
\]
together with the Jacobi triple product
\[
f(a,b)=(-a;ab)_\infty(-b;ab)_\infty(ab;ab)_\infty.
\]
The argument also uses classical \(2\)-dissections, \(5\)-dissections of \(E_1\) and \(1/E_1\), product identities for the Rogers–Ramanujan continued fraction
\[
T(q):=q^{1/5}R(q)=\frac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty},
\]
and theta-function addition formulas. Extracting the \(q^{5n+2}\) terms and reducing modulo \(5\) yields
\[
a_{1,10}(10n+5)\equiv 0\pmod 5,\qquad a_{3,10}(10n+5)\equiv 0\pmod 5.
\]

The significance stated for these proofs is twofold. First, they provide exact \(q\)-product formulas for arithmetic progressions of mock theta coefficients, enabling congruence proofs in the style of Ramanujan’s partition congruences. Second, they extend the arithmetic theory of Appell–Lerch sums by exhibiting hidden modular structure in coefficient sequences that are not modular forms themselves but are closely related to mock modular forms [1807.03957].

## 4. Necklace–restricted multiset formulation

A distinct conjecture of S. H. Chan concerns algebraic combinatorics. The 2025 paper titled “A Bijection between Necklaces and Restricted Multisets” states that it proves Chan’s conjecture establishing a bijection between the set of necklaces of length \(n\) with at most \(q\) colors and the set of periodic functions
\[
f:\mathbb Z_n\to\{0,1,\dots,q-1\}
\]
whose weighted sum is divisible by \(n\), where \(q\) and \(n\) are coprime positive integers [2507.19940].

In the accompanying description, a necklace of length \(n\) is interpreted as a cyclic word modulo rotation, and a periodic function \(f:\mathbb Z_n\to\{0,1,\dots,q-1\}\) is interpreted as an \(n\)-periodic sequence. The weighted-sum condition is described by
\[
\sum_{i=0}^{n-1} i\,f(i)\equiv 0 \pmod n.
\]
The same description also gives the multiset interpretation: \(f\) may be viewed as a multiset on \(\mathbb Z_n\) in which the element \(i\) appears with multiplicity \(f(i)\), bounded by \(q-1\). In this formulation, the divisibility condition becomes
\[
\sum_{i\in \mathbb Z_n} i\,m_i \equiv 0 \pmod n.
\]

This conjecture is qualitatively different from Chan’s Appell–Lerch congruences. It is bijective rather than divisibility-theoretic, and it links orbit spaces of cyclic group actions with modular weighted-sum constraints on bounded-multiplicity multisets.

## 5. Canonical representatives, coprimality, and orbit structure

The description attached to the necklace theorem presents a standard mechanism behind the bijection. One introduces the weighted-sum statistic
\[
W(f)=\sum_{i=0}^{n-1} i\,f(i)\pmod n,
\]
regards a necklace as an orbit of words under cyclic rotation, and then selects a canonical representative by imposing the normalization
\[
W(f)\equiv 0 \pmod n.
\]
This suggests an orbit-normalization principle: each cyclic orbit should contain a unique representative satisfying the divisibility condition [2507.19940].

The arithmetic condition
\[
\gcd(n,q)=1
\]
is described as essential. The same description states that the coprimality is what makes the modular normalization work, because it guarantees the existence and uniqueness of the required rotation. In this sense, Chan’s conjecture is not merely an equinumerosity statement. Its content is the existence of a natural bijective representative-selection rule compatible with cyclic symmetry and modular weighted sums.

A plausible implication is that the theorem gives a combinatorial explanation for why necklaces are equinumerous with the weighted-sum-zero functions. In the reconstructed framework attached to the paper, this is the conceptual reason the conjecture is expressed as a bijection rather than as a counting identity alone.

## 6. Ambiguity of the label and related prompt-attributed usages

The label “Swee Hong Chan’s conjecture” is not used uniformly across the supplied arXiv material. Some papers are directly about conjectures of S. H. Chan, while others are linked only by attribution in the prompt or by a conceptual analogy.

In “Proofs of Some Conjectures of Chan on Appell-Lerch Sums” the attribution is explicit and unambiguous: the paper proves conjectured congruences of Chan for coefficients of sixth order mock theta functions and generalized Appell–Lerch sums [1807.03957]. In “A Bijection between Necklaces and Restricted Multisets,” the abstract likewise explicitly presents a proof of Swee Hong Chan’s conjecture in the combinatorial setting [2507.19940].

By contrast, “A Conjecture on Induced Subgraphs of Cayley Graphs” states a conjecture of Potechin–Tsang: for any Cayley graph \(X=\Gamma(G,S)\) and any subset \(U\subseteq G\) with \(|U|>|G|/2\), the induced subgraph \(X(U)\) has maximum degree at least \(\sqrt{|S|/2}\). The supplied description says that this conjecture is “attributed in the prompt to Swee Hong Chan’s conjecture,” while the paper itself proves the statement for abelian groups with the stronger bound
\[
\Delta(X(U))\ge \sqrt{\frac{|S|+t}{2}},
\]
where \(t\) is the number of elements of \(S\) of order \(2\) [2003.13166].

Similarly, “Proof of the Generalization of the Sawayama-Thébault Theorem” is described as proving two conjectures posed in 2016, but the supplied summary states that the paper itself does not explicitly name Swee Hong Chan. It proves a generalized Sawayama lemma and a generalized Sawayama–Thébault theorem in a Laguerre-geometric framework, with the fixed point identified as the incenter \(I\) of triangle \(ABC\) [2603.18651].

Finally, “Phase shifts vs time delays: Sagnac and Hong-Ou-Mandel” does not formulate a conjecture of Chan. The supplied description instead presents it as offering a conceptual link: the paper argues that the Sagnac effect is fundamentally a time delay
\[
\Delta\tau_S=\frac{4A\Omega}{c^2},
\]
and that Hong–Ou–Mandel interference is insensitive to constant phase shifts but sensitive to time delays, thereby operationalizing the timing-based character of the effect [1104.0970].

The most stable usage of the phrase therefore remains the pair of Chan-originating problems in \(q\)-series arithmetic and bijective combinatorics. In those settings, “Swee Hong Chan’s conjecture” refers to precise statements that have now been proved: explicit congruences for mock-theta/Appell–Lerch coefficients and a bijection between necklaces and restricted multisets under a coprimality hypothesis.

Source: https://www.emergentmind.com/topics/swee-hong-chan-s-conjecture