Finite parameter tests for partition congruences and the surviving classes of an elongated partition conjecture
Henry Zweiman
September 8, 2026
Abstract
We prove the seven remaining congruence families in a conjecture of Guadalupe on elongated plane partitions, and determine exactly when its final pair of congruences holds. The two progressions conjectured to vanish modulo vanish identically precisely for parameter classes ; they vanish modulo for every . The obstruction at the initial coefficients was identified by Patel. We prove the converse and the other families using a finite parameter principle and explicit Radu certificates. The principle reduces a parameter family modulo to initial parameter values when its normalized parameter ratio is modulo . We give the complete certificate data and exact arithmetic verifiers.
1. Introduction and results
For , write . The elongated plane partition functions considered by Andrews and Paule, and subsequently by several authors, have generating functions
Guadalupe [1] proved numerous congruences modulo small powers of and stated eight further families in Conjecture 7.1. A generic transfer theorem quoted there changes by a substantially larger power of than the step appearing in the conjecture. Controlling this smaller parameter step is therefore essential.
The last family of that conjecture is false. Patel’s posted counterexample [2] proves
These formulas rule out three parameter classes. Our main result proves that the two remaining classes work for every coefficient, and establishes the other seven families. We include an independently checked proof of (2) for completeness; we make no priority claim for that obstruction.
Theorem 1.
For every , the congruences in Table 1 hold. A list of residues means that the assertion holds separately at each listed residue.
Table 1. Uniform congruences.
Parameter
Argument of
Dividing modulus
Moreover, for a fixed , each of the following assertions is equivalent to or :
The result uses a standard modular finite coefficient criterion, with explicitly bounded exact computations. It does not use numerical evidence as a substitute for an infinite argument. All infinite parameter dependence is handled by the following elementary principle.
2. A finite parameter principle
For and , let denote the additive subgroup of consisting of series whose coefficients in degrees congruent to modulo are divisible by . Multiplication by any series in preserves this subgroup.
Lemma 2.
Let be prime, let , and let . Suppose that and
Put . If for , then for every .
Proof.
The inverse also belongs to . Thus the hypotheses imply for . The integer binomial identity gives
for . For , the entire left side is divisible by . Expanding in powers of proves that it belongs to the subgroup. Multiplication by completes the proof.
For any integer unit series , reduction modulo gives . This identity holds for infinite formal series because each coefficient depends on only finitely many factors and terms.
Corollary 3.
Fix . For , to prove
it is enough to prove the assertion for every at . For , to prove
it is enough to prove the assertion for every at .
Proof.
In the first case use , and in Lemma 2, with . In the second case use and . If , then . Hence and .
3. A modular certificate with explicit cusp bounds
Write and . The orbit consists of the least nonnegative residues
Unit squares modulo are modulo , so this is integral.
The admissibility conditions for are the following: every prime divisor of divides ; every with divides ; and
There is an additional condition for even , irrelevant here. We use only odd .
For and an integer vector , put
If the tuple is admissible and for a complete set of cusp representatives, the criterion states that the congruences
hold for every provided they hold for , where
Here is any positive integer. In particular this criterion applies directly to .
Proposition 4.
Let , , , , and . Suppose . Set
where the coordinates are indexed by . Define and
If for every and , then these congruences hold for all .
Proof.
The binomial congruence shows that . It follows, for example, by induction from , then multiplication of formal series. Inverting a unit series preserves a coefficientwise congruence.
Take . The index is and . Also . The first four admissibility conditions are immediate. The last follows because is either or and hence divides .
Since is squarefree, the four matrices , , represent the cusps. Directly from the definition,
We justify these equalities to make the cusp bounds explicit. For , put . If , the scaled sum is , minimized at . If , it is
since and . At , the integer is odd; the corresponding expressions are for and a positive expression for . Both extremizing gcd values occur. At , the integer is coprime to , giving the last two values in (9).
The most negative entry is the first. Thus has and satisfies every cusp inequality. Substitution into (6) gives (8), proving the proposition.
4. Base cases and sufficiency
Table 2 gives every base case used. In each row, , are given by (7), and the final column is . Formula (5) gives exactly the displayed set . All rows satisfy for .
The supplied exact arithmetic verification checks all coefficients required by the table. The largest index is . Every tested residue is zero modulo the stated modulus. Proposition 4 therefore proves each base congruence for all .
Table 2. Complete base certificates.
58
25
3
165
183
25
3
424
308
25
3
683
83
125
3
763
208
125
3
1907
333
125
3
3051
100
125
3
962
225
125
3
2106
350
125
3
3250
5
125
3
114
130
125
3
1258
255
125
3
2402
30
125
3
297
155
125
3
1442
280
125
3
2586
60
125
3
593
60
125
3
593
185
125
3
1738
185
125
3
1738
310
125
3
2882
310
125
3
2881
58
125
4
803
58
125
4
803
183
125
4
1723
183
125
4
1723
308
125
4
3017
308
125
4
3017
433
125
4
4311
433
125
4
4312
58
125
5
2303
683
125
5
6900
1308
125
5
13371
308
125
5
4142
933
125
5
8739
1558
125
5
15210
For the first six rows of Table 1, the certificate table supplies . Corollary 3 proves the desired congruences for every . For the two rows modulo , it supplies , so the same corollary again applies.
For the stronger congruences, the table supplies
The second part of Corollary 3 yields both progressions modulo for and . These are exactly the parameter classes in Theorem 1. This proves sufficiency.
5. The obstruction and necessity
We record Patel’s initial-coefficient argument [2], with an independent integer verification. Put . Then . Below degree , the factor is . Consequently each of
is, modulo , an integer linear combination of for . Indeed, expand and discard for .
The five exact initial evaluations are
The evaluation matrix is unit lower triangular over . These values therefore establish (2) for every , since . No division by a nonunit is involved.
For reproducibility, the evaluations can be made using only integers:
This follows by formal logarithmic differentiation of (1). The supplied independent verifier uses exact division in and checks its divisibility at every step. Either of (3) and (4), at , forces . Together with the sufficiency already proved, this completes Theorem 1.
6. Verification and reproducibility
The supplementary files contain one authoritative manifest of the base cases, the complete rational certificates, and two coefficient algorithms. The certificate generator computes every orbit and cusp minimum with integers and rational numbers, and also checks the closed formulas (9). Its record includes , the largest coefficient index and the number of zero checks.
The main verifier forms Euler’s pentagonal series in , inverts its constant coefficient , and computes in that ring. Truncated multiplication, inversion and exponentiation give precisely the first coefficients of the infinite formal series. The implementation uses Python with python-flint 0.9.0; its mathematical arithmetic is exact. The separate integer verifier evaluates (10) through degree .
The mathematical dependence on computation is limited and explicit: the finite modular zero checks in Table 2 and the five initial evaluations. The finite criterion, its cusp hypotheses, the parameter extension, and the necessity argument are proved or cited above. The supplementary code and machine-readable records are part of the manuscript’s reproducibility package.
References
[1] R. Guadalupe, The -elongated plane partition function modulo small powers of , arXiv:2504.08627v2 (2025), arXiv record.
[2] S. Patel, Counterexample, solution posted to “Conjectural congruences for the -elongated plane partition function modulo powers of ,” MathDB, accessed September 8, 2026, online solution.
[3] L. Wang, Arithmetic properties of -regular bipartitions, Bull. Aust. Math. Soc. 95 (2017), 353–364, doi:10.1017/S0004972716000964.
[4] D. Tang, New congruences for broken -diamond partitions, J. Integer Seq. 21 (2018), Article 18.5.8, publisher PDF.
[5] S. Radu, An algorithmic approach to Ramanujan’s congruences, Ramanujan J. 20 (2009), 215–251.