Ideal Degree-3 Prouhet–Tarry–Escott Problem

• The paper presents the ideal degree-three Prouhet–Tarry–Escott problem as the minimal cubic equal-sums-of-like-powers case, requiring two distinct 4-element multisets with matching power sums for k=1,2,3.
• It reformulates the problem as a quartic difference equated to a constant, using affine equivalence and symmetric properties to simplify identification of invariants.
• The topic bridges classical number theory with applications in quantum field theory, spectral theory, and complexity through structured search strategies and explicit polynomial parametrizations.

The ideal degree-three Prouhet–Tarry–Escott problem is the minimal nontrivial cubic equal-sums-of-like-powers problem: one seeks two distinct 4-element multisets of integers,

$$A=\{a_1,a_2,a_3,a_4\},\qquad B=\{b_1,b_2,b_3,b_4\},$$

such that

$$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$

It is the $k=3$ instance of the ideal Prouhet–Tarry–Escott problem because the ideal case is the maximal case $m=n-1$, equivalently the minimal-size case $n=k+1$; here $n=4$ (Coppersmith et al., 2023, Choudhry, 2016).

1. Canonical formulation

In the standard two-set formulation, the Prouhet–Tarry–Escott problem asks for two distinct multisets of the same size whose power sums agree through a prescribed degree. For degree $3$, the ideal case is therefore exactly the 4-vs-4 system

$a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$

$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$

$a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$

A nontrivial solution must satisfy $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$0, so size $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$1 is minimal for degree $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$2 (Choudhry, 2016).

The natural equivalence relation is affine equivalence together with the obvious multiset symmetries. If $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$3, then for any nontrivial affine transformation $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$4, the transformed multisets are again a solution; permutations within each multiset and interchange of the two sides are likewise harmless (Choudhry, 2016, Coppersmith et al., 2023). This is the framework in which degree-three solutions are usually normalized.

A useful historical precursor is the Euler–Goldbach identity

$$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$5

which is a size-4 solution of degree $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$6, not of the ideal degree-three problem (Coppersmith et al., 2023).

2. Quartic reformulation and the constant $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$7

For ideal solutions of size $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$8, the Prouhet–Tarry–Escott conditions are equivalent to a polynomial-difference condition: $$\sum_{i=1}^4 a_i^k=\sum_{i=1}^4 b_i^k,\qquad k=1,2,3.$$9 In the degree-three case $k=3$0, this becomes

$k=3$1

where $k=3$2 is a constant independent of $k=3$3 (Coppersmith et al., 2023).

Expanding the two monic quartics,

$k=3$4

$k=3$5

shows that the quartics differ by a constant exactly when

$k=3$6

By Newton’s identities, this is equivalent to equality of the first three power sums (Coppersmith et al., 2023, Shuwen, 13 Jun 2025).

The constant has several useful exact formulas. In the survey formulation,

$k=3$7

and for every $k=3$8,

$k=3$9

These identities make the constant a practical invariant in both structural arguments and computer searches (Shuwen, 13 Jun 2025).

Over $m=n-1$0, the size-4 constant satisfies strong divisibility constraints. Table 1 of the 2023 search paper gives

$m=n-1$1

so every ideal degree-three integer solution has constant divisible by $m=n-1$2 (Coppersmith et al., 2023). This sharpens the general divisibility statement $m=n-1$3 in the case $m=n-1$4.

3. Symmetry and reduction to sums of two squares

For even $m=n-1$5, a symmetric Prouhet–Tarry–Escott solution is one with

$m=n-1$6

In the degree-three ideal case this means

$m=n-1$7

possibly with multiplicities (Coppersmith et al., 2023). In that form, the first and third moments vanish automatically on each side, so the only nontrivial condition is

$m=n-1$8

Recent work refines this symmetric description by allowing a common center $m=n-1$9. A symmetric ideal degree-three solution is one for which both $n=k+1$0 and $n=k+1$1 are invariant under sign reversal. After centering and doubling,

$n=k+1$2

with $n=k+1$3 all of the same parity, and the full degree-three system is equivalent to the single equation

$n=k+1$4

(Tsai et al., 5 Jun 2026). This identifies the symmetric locus with the arithmetic of representations as sums of two squares.

That reduction supports an asymptotic count. If $n=k+1$5 denotes the number of nontrivial symmetric integer solutions of height at most $n=k+1$6, counted with unordered multiset conventions and summed over admissible centers, then

$n=k+1$7

The logarithmic factor comes from the second moment of the sum-of-two-squares representation function (Tsai et al., 5 Jun 2026).

4. Explicit families and representative quartets

A modern explicit polynomial parametrization is given by Choudhry’s degree-three theorem. With four arbitrary parameters $n=k+1$8,

$n=k+1$9

$n=4$0

where

$n=4$1

This yields an ideal degree-three solution, and the common sums $n=4$2 are symmetric functions of $n=4$3 (Choudhry, 2021).

Representative explicit quartets recorded in the recent literature include the following.

Type	Quartets	Source
Parametric family	$n=4$4 from $n=4$5	(Choudhry, 2021)
Symmetric example	$n=4$6	(Lee et al., 12 Mar 2026)
Example with repetition	$n=4$7	(Lee et al., 12 Mar 2026)
Multigrade chain	$n=4$8	(Shuwen, 13 Jun 2025)

The pair $n=4$9 and $3$0 is symmetric about $3$1; after subtracting $3$2 and doubling, it becomes

$3$3

and

$3$4

(Tsai et al., 5 Jun 2026). The example $3$5 is singled out in the physics literature as the solution minimizing $3$6 among the listed charge assignments (Lee et al., 12 Mar 2026).

The 2025 survey also records a trigonometric symmetric identity and several huge integer ideal quartets, including both symmetric and non-symmetric examples (Shuwen, 13 Jun 2025).

5. Larger Prouhet constructions and computational search

Ideal quartets sit inside a larger family of non-ideal degree-three constructions. The classical binary Prouhet partition for degree $3$7 is

$3$8

which satisfies equality of sums of powers for $3$9 but uses $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$0 terms on each side, not the ideal $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$1 (Nguyen, 2014). The same 8-vs-8 cubic partition is the $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$2 case of the generalized digit-map construction (Ma, 14 Sep 2025).

Combinatorial Prouhet constructions yield the same scale. The binary generalized Thue–Morse word

$a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$3

encodes the partition

$a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$4

and the cited work reports that $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$5 contains just one word, namely this length-16 Thue–Morse prefix (Bolker et al., 2013).

The 2018 digit-sum paper shows how cancellations can beat the classical size barrier. For $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$6, $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$7, $a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$8, the generalized construction produces

$a_1+a_2+a_3+a_4=b_1+b_2+b_3+b_4,$9

with equal first and second power sums; cancelling the common value $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$0 yields the smaller partition

$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$1

and the paper explicitly remarks that its results are amenable to a computational search, which may discover new, smaller, solutions to the classical problem (Wakhare et al., 2018). This suggests a natural search strategy for degree three: start from a structured 8-vs-8 cubic identity and maximize cross-cancellation.

Recent survey and search papers formalize complementary search heuristics. For size $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$2, if $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$3, then modulo $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$4 the two multisets coincide after reordering, giving obligatory local constraints (Coppersmith et al., 2023). The 2025 survey also normalizes ideal quartets by

$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$5

and records the interlacing pattern

$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$6

together with

$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$7

(Shuwen, 13 Jun 2025). These conditions are designed to prune searches for normalized integer quartets.

6. Extensions, applications, and research status

A 2016 construction paper states that the complete ideal solution of the Tarry–Escott problem is known only when $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$8 or $a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2+b_4^2,$9, and omits the cubic formulas because they were already known (Choudhry, 2016). By contrast, a 2023 existence paper proves that an ideal solution of the Prouhet–Tarry–Escott problem with any degree always exists, but the proof is representation-theoretic and does not exhibit explicit cubic tuples (Sun et al., 2023).

Beyond $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$0, the degree-three ideal problem changes character. Over the Gaussian integers,

$a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$1

and the associated quartic difference is

$a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$2

so the constant is a unit (Caley, 2010). This is a genuinely Gaussian ideal degree-three solution.

The cubic ideal problem also appears in other areas. In one quantum-field-theoretic model, the anomaly-cancellation conditions become exactly

$a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$3

so the lower bound $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$4 implies that at least four states are required in the minimal case (Lee et al., 12 Mar 2026). In weighted-path spectral theory, the existence of a $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$5-chain with periodic cospectral vertices at positions $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$6 and $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$7 is equivalent to a solution of $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$8, i.e. a disjoint ideal degree-three solution (Cançado et al., 12 Sep 2025). In complexity theory, explicit small PTE constructions form a barrier in reductions from Moments Subset Sum to Reed–Solomon decoding, and the absence of explicit solutions of size $a_1^3+a_2^3+a_3^3+a_4^3=b_1^3+b_2^3+b_3^3+b_4^3.$9 is identified as the main obstruction to extending those hardness results (Gandikota et al., 2016).

Taken together, these results place the ideal degree-three Prouhet–Tarry–Escott problem in a distinctive position. It is a classical low-degree case for which complete ideal solutions are already known in the older literature (Choudhry, 2016), but it remains a live object in current research because of its quartic algebra, its large symmetric locus, its search-theoretic structure, and its unexpectedly broad connections to number fields, spectral theory, quantum field theory, and complexity theory (Tsai et al., 5 Jun 2026).