Universal Quaternary Sum

• Universal quaternary sum is a four-term representation problem uniting diverse arithmetic settings such as quadratic forms, polygonal numbers, and quaternion rings.
• Techniques like escalator trees, quadratic reductions, and Ramanujan theta series analyses provide finite test criteria for establishing universality.
• The theory bridges classical results like Lagrange’s four-square theorem with modern studies in quaternion rings and polynomial optimization, highlighting wide-ranging applications.

A universal quaternary sum is a four-term or four-variable expression that represents every element of a specified arithmetic domain. The phrase is not uniform across the literature: in classical quadratic-form theory it refers to positive-definite quaternary forms representing all positive integers; in the theory of polygonal numbers it denotes four-variable sums of $P_m(x)$ representing all nonnegative integers; in quaternion rings it means that every element of the subgroup generated by squares is a sum of four quaternion squares; and over other rings, such as $\mathbb{Z}[i]$ or rings of integers in real quadratic fields, it refers to universality relative to Gaussian integers or totally positive algebraic integers (Yang, 2023, Ju et al., 2017, Sidokhine, 2013, Cooke et al., 2016, Thompson, 2016).

1. Terminological scope

The common structural feature is a representation problem with four summands. In the polygonal-number setting, a quaternary sum means a function

$$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$

and universality means that $F$ represents every positive integer or every nonnegative integer, depending on the convention of the paper (Yang, 2023, Ju et al., 2018, Ju et al., 2017). In quaternionic Waring-type problems, the relevant condition is $g_{a,b}(2)=4$, where $g_{a,b}(2)$ is the minimal number of quaternion squares needed to represent every element of

$Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$

the additive subgroup generated by squares in $Q_{a,b}$ (Cooke et al., 2016).

Context	Quaternary object	Universality target
Mixed polygonal sums	A 4-variable sum of $P_m$-terms	Every positive or nonnegative integer (Yang, 2023, Ju et al., 2018)
Quaternion rings	Four quaternion squares	Every element of $Q'_{a,b}$ (Cooke et al., 2016)
Other rings and fields	A quaternary quadratic form	Every Gaussian integer or every totally positive element (Sidokhine, 2013, Thompson, 2016)

This multiplicity of meanings explains why the term is best read relative to its ambient category. A universal quaternary sum is therefore not a single object class, but a family of representation phenomena linked by the number four and by full representability.

2. Classical universality and finite test sets

The archetype is Lagrange’s four-square theorem, which asserts that every positive integer is a sum of four integer squares. Several of the cited works explicitly frame later developments as analogues or extensions of this quaternary universality paradigm (Cooke et al., 2016, Thompson, 2016). In the theory of integral quadratic forms, this is complemented by the Conway–Schneeberger Fifteen Theorem and the Bhargava–Hanke 290-Theorem, both of which are invoked as models for finite universality criteria (Yang, 2023, Ju et al., 2018, Ju et al., 2017).

For sums of triangular numbers and squares, universality admits an exact finite characterization. A sum in the class $\mathbb{Z}[i]$0, equivalently a sum of triangular numbers and squares, is universal if and only if it represents

$\mathbb{Z}[i]$1

and each of these 22 integers is critical; consequently $\mathbb{Z}[i]$2 (Yang, 2023). In this setting, quaternary sums are the depth-$\mathbb{Z}[i]$3 nodes of the escalator tree, and the paper shows that all depth-$\mathbb{Z}[i]$4 nodes are universal except an explicit finite list (Yang, 2023).

A related but broader asymptotic statement appears for sums of generalized $\mathbb{Z}[i]$5-gonal numbers. If $\mathbb{Z}[i]$6 is the largest truant appearing in the escalator tree, then representing every integer up to $\mathbb{Z}[i]$7 suffices for universality, and for every $\mathbb{Z}[i]$8,

$\mathbb{Z}[i]$9

The same work records exact values $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$0, $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$1, $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$2, $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$3, and $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$4 (Kane et al., 2018). This suggests that quaternary universality sits inside a wider finite-check framework, even when a full classification is unavailable.

3. Mixed polygonal-number realizations

The polygonal-number literature contains the largest explicit catalogues of universal quaternary sums. For generalized $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$5- and $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$6-gonal numbers, a mixed sum

$$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$7

is universal if it represents every nonnegative integer, and a quaternary sum means $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$8 (Ju et al., 2018). In this framework there are exactly $$F:\mathbb{Z}^4\to\mathbb{Z},\qquad F(x_1,x_2,x_3,x_4)=a_1P_{m_1}(x_1)+a_2P_{m_2}(x_2)+a_3P_{m_3}(x_3)+a_4P_{m_4}(x_4),$$9 proper quaternary universal mixed sums, and across all ranks there are exactly $F$0 proper universal mixed sums of generalized $F$1- and $F$2-gonal numbers (Ju et al., 2018). The same paper proves the “61-theorem”: such a mixed sum is universal if and only if it represents

$F$3

(Ju et al., 2018).

For generalized octagonal numbers, the quaternary sum

$F$4

is universal if $F$5 has an integer solution for every positive integer $F$6 (Ju et al., 2017). One paper completes Sun’s list of unresolved quaternary octagonal cases by proving universality for $F$7, and then establishes the “octagonal theorem of sixty”: an arbitrary sum of generalized octagonal numbers is universal if and only if it represents $F$8 (Ju et al., 2017).

Recent work extends this to genuinely mixed quaternary sums involving generalized $F$9-, $g_{a,b}(2)=4$0-, $g_{a,b}(2)=4$1-, and $g_{a,b}(2)=4$2-gonal numbers. A 2025 paper determines the universality of many quaternary sums

$g_{a,b}(2)=4$3

using products of Ramanujan’s theta functions (Bulkhali et al., 18 Jul 2025). The concrete lists in that paper include universal families of the forms $g_{a,b}(2)=4$4, $g_{a,b}(2)=4$5, $g_{a,b}(2)=4$6, $g_{a,b}(2)=4$7, and $g_{a,b}(2)=4$8, among others (Bulkhali et al., 18 Jul 2025). A plausible implication is that theta-function factorizations are becoming a unifying technique for quaternary mixed polygonal sums that were previously treated by ad hoc ternary-form arguments.

4. Quaternion rings and universality beyond $g_{a,b}(2)=4$9

A distinct use of the term appears in quaternion rings

$g_{a,b}(2)$0

where every square has even pure coefficients and the additive subgroup generated by squares is

$g_{a,b}(2)$1

The quantity $g_{a,b}(2)$2 is the minimal number of squares needed to represent every element of $g_{a,b}(2)$3, and the paper states explicitly that if $g_{a,b}(2)$4, this is a universal quaternary sum in $g_{a,b}(2)$5 (Cooke et al., 2016). The central theorem is

$g_{a,b}(2)$6

and each of $g_{a,b}(2)$7 occurs for infinitely many pairs $g_{a,b}(2)$8 (Cooke et al., 2016). Thus four squares are universal in some quaternion rings but not uniformly: for example, $g_{a,b}(2)$9 for all $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$0, whereas $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$1 for all $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$2 (Cooke et al., 2016). This directly contradicts any naive expectation that the integer four-square phenomenon should persist unchanged in noncommutative settings.

Over Gaussian integers, universality can also be quaternary but no longer integer-valued. The form

$Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$3

is universal over $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$4, meaning that for every $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$5 there exist Gaussian integers $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$6 with

$Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$7

(Sidokhine, 2013). The proof combines a classification of Gaussian primes by norm modulo $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$8, a Gaussian version of Fermat’s little theorem in $Q'_{a,b}=\{x_0+2x_1i+2x_2j+2x_3k\},$9, the Mordell–Niven theorem, and a generalized Euler identity showing multiplicative closure of the value set (Sidokhine, 2013).

For real quadratic number fields, the analogue behaves much more rigidly. The quaternary form

$Q_{a,b}$0

over $Q_{a,b}$1 is universal only for $Q_{a,b}$2; for every other real quadratic field the sum of four squares is not universal (Thompson, 2016). Over $Q_{a,b}$3 the associated Hilbert theta series has zero cusp part at level $Q_{a,b}$4, yielding an explicit Jacobi-type formula for the representation numbers $Q_{a,b}$5 (Thompson, 2016). This suggests that quaternary universality over number fields depends as much on the ambient field and its modular forms as on the formal appearance of the sum.

5. Primitive universality and sharper quaternary classifications

Universality does not imply primitive universality. For a positive definite integral quaternary quadratic form $Q_{a,b}$6, primitive universality means that every positive integer is represented by some vector $Q_{a,b}$7 with $Q_{a,b}$8 (Ju et al., 2022). One paper determines this refinement completely within the Conway–Schneeberger universe of quaternary forms: there are exactly $Q_{a,b}$9 equivalence classes of universal quaternary quadratic forms, exactly $P_m$0 equivalence classes of primitively almost universal quaternary quadratic forms, and exactly $P_m$1 equivalence classes of primitively universal quaternary quadratic forms (Ju et al., 2022). For the remaining $P_m$2 primitively almost universal but not primitively universal classes, the set of positive integers not primitively represented is determined explicitly (Ju et al., 2022).

This result changes the interpretation of universal quaternary sums in the classical quadratic-form setting. A universal quaternary form may represent every positive integer, yet fail to do so primitively for finitely many integers. The paper’s analysis is lattice-theoretic: it uses $P_m$3-adic primitive universality, the $P_m$4-transformation, ternary core sublattices, class number $P_m$5 arguments, and precise congruence control on representations by cores such as $P_m$6, $P_m$7, and $P_m$8 (Ju et al., 2022). A common misconception is therefore that quaternary universality is a single invariant. The primitive classification shows that there are at least three relevant layers: universality, primitive almost universality, and primitive universality.

An adjacent refinement appears for sums of generalized pentagonal numbers. A ternary sum

$P_m$9

can be almost universal without being universal, and the obstruction is described in terms of primitive spinor exceptions of an associated ternary lattice $Q'_{a,b}$0 (Wu et al., 2020). This is not itself a quaternary theory, but it clarifies a mechanism that often disappears or weakens in quaternary settings: after rank $Q'_{a,b}$1, local-global regularity becomes substantially stronger.

6. Methods, modularity, and adjacent meanings

Three proof architectures recur across the literature. The first is the escalator tree, inherited from Bhargava’s method and used heavily for mixed polygonal sums: nodes are partial sums, the truant is the least missing integer, and new variables are added until universality is reached (Yang, 2023, Ju et al., 2018, Kane et al., 2018). The second is the reduction of polygonal-number sums to quadratic forms with congruence conditions, followed by genus theory, class-number-$Q'_{a,b}$2 arguments, or regular ternary-form analysis (Yang, 2023, Ju et al., 2017, Wu et al., 2017). The third is analytic: theta-series decompositions

$Q'_{a,b}$3

local densities, Eisenstein lower bounds, cusp-form estimates, and Ramanujan theta identities all function as representation-theoretic engines (Thompson, 2016, Yang, 2023, Bulkhali et al., 2024, Bulkhali et al., 18 Jul 2025).

Ramanujan’s theta functions are especially prominent in the most recent quaternary polygonal literature. The specializations

$Q'_{a,b}$4

encode squares, triangular numbers, generalized pentagonal numbers, and generalized octagonal numbers, respectively, and product identities between them transfer universality from one quaternary sum to another (Bulkhali et al., 2024, Bulkhali et al., 18 Jul 2025). This suggests a shift from isolated classifications toward a transformation theory of universality.

A distinct but mathematically adjacent usage of quaternary universality appears in polynomial optimization. For any non-negative 4-ary quartic form $Q'_{a,b}$5, there exist non-negative quadrics $Q'_{a,b}$6 and $Q'_{a,b}$7 such that $Q'_{a,b}$8 is a sum of squares of quartics (Pasechnik, 2015). This is not a universal quaternary sum in the additive-number-theoretic sense, but it preserves the same four-variable geometry and the same emphasis on universal representation by structured quadratic data. It therefore marks a boundary of the term rather than a central instance of it.

Across these settings, the number four is neither merely historical nor merely notational. In some domains it is the exact universal threshold, as in classical four squares and certain polygonal or Gaussian examples (Sidokhine, 2013, Ju et al., 2017). In others it is only one value among several possibilities, as in quaternion rings where the minimal number can be $Q'_{a,b}$9, $\mathbb{Z}[i]$00, or $\mathbb{Z}[i]$01 (Cooke et al., 2016). The modern theory of universal quaternary sums is accordingly best understood not as a single theorem, but as a network of representation problems in which four summands interact with local arithmetic, modular forms, and the structure of the ambient ring.