---
title: Ideal Degree-3 Prouhet–Tarry–Escott Problem
url: https://www.emergentmind.com/topics/ideal-degree-three-prouhet-tarry-escott-problem
type: topic
---

# Ideal Degree-3 Prouhet–Tarry–Escott Problem

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 \(n=4\) [2304.11254][1603.00206].

## 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 \(n\ge k+1\), so size \(4\) is minimal for degree \(3\) [1603.00206].

The natural equivalence relation is affine equivalence together with the obvious multiset symmetries. If \(A=_k B\), then for any nontrivial affine transformation \(x\mapsto Mx+K\), the transformed multisets are again a solution; permutations within each multiset and interchange of the two sides are likewise harmless [1603.00206][2304.11254]. This is the framework in which degree-three solutions are usually normalized.

A useful historical precursor is the Euler–Goldbach identity
\[
\{a,b,c,a+b+c\}=_2\{0,a+b,a+c,b+c\},
\]
which is a size-4 solution of degree \(2\), not of the ideal degree-three problem [2304.11254].

## 2. Quartic reformulation and the constant \(C_4\)

For ideal solutions of size \(n\), the Prouhet–Tarry–Escott conditions are equivalent to a polynomial-difference condition:
\[
\deg\!\left(\prod_{i=1}^n (x-a_i)-\prod_{i=1}^n (x-b_i)\right)<1.
\]
In the degree-three case \(n=4\), this becomes
\[
\prod_{i=1}^4 (x-a_i)-\prod_{i=1}^4 (x-b_i)=C_4,
\]
where \(C_4\) is a constant independent of \(x\) [2304.11254].

Expanding the two monic quartics,
\[
\prod_{i=1}^4 (z-a_i)=z^4-e_1(A)z^3+e_2(A)z^2-e_3(A)z+e_4(A),
\]
\[
\prod_{i=1}^4 (z-b_i)=z^4-e_1(B)z^3+e_2(B)z^2-e_3(B)z+e_4(B),
\]
shows that the quartics differ by a constant exactly when
\[
e_1(A)=e_1(B),\qquad e_2(A)=e_2(B),\qquad e_3(A)=e_3(B).
\]
By Newton’s identities, this is equivalent to equality of the first three power sums [2304.11254][2506.11429].

The constant has several useful exact formulas. In the survey formulation,
\[
C=a_1a_2a_3a_4-b_1b_2b_3b_4,
\]
and for every \(j\),
\[
-C=\prod_{i=1}^{4}(a_j-b_i),\qquad C=\prod_{i=1}^{4}(b_j-a_i).
\]
These identities make the constant a practical invariant in both structural arguments and computer searches [2506.11429].

Over \(\mathbb Z\), the size-4 constant satisfies strong divisibility constraints. Table 1 of the 2023 search paper gives
\[
C_4:\;2^2\cdot 3^2,
\]
so every ideal degree-three integer solution has constant divisible by \(36\) [2304.11254]. This sharpens the general divisibility statement \((n-1)!\mid C_n\) in the case \(n=4\).

## 3. Symmetry and reduction to sums of two squares

For even \(n\), a symmetric Prouhet–Tarry–Escott solution is one with
\[
A=-A,\qquad B=-B.
\]
In the degree-three ideal case this means
\[
A=\{\pm u,\pm v\},\qquad B=\{\pm r,\pm s\},
\]
possibly with multiplicities [2304.11254]. In that form, the first and third moments vanish automatically on each side, so the only nontrivial condition is
\[
u^2+v^2=r^2+s^2.
\]

Recent work refines this symmetric description by allowing a common center \(c\in \tfrac12\mathbb Z\). A symmetric ideal degree-three solution is one for which both \(A-c\) and \(B-c\) are invariant under sign reversal. After centering and doubling,
\[
2(A-c)=\{\pm x,\pm y\},\qquad 2(B-c)=\{\pm u,\pm v\},
\]
with \(x,y,u,v\in\mathbb Z\) all of the same parity, and the full degree-three system is equivalent to the single equation
\[
x^2+y^2=u^2+v^2
\]
[2606.07735]. This identifies the symmetric locus with the arithmetic of representations as sums of two squares.

That reduction supports an asymptotic count. If \(N_{\mathrm{sym}}(H)\) denotes the number of nontrivial symmetric integer solutions of height at most \(H\), counted with unordered multiset conventions and summed over admissible centers, then
\[
N_{\mathrm{sym}}(H)=\frac{4\log 2}{3\pi^2}H^3\log H+O(H^3).
\]
The logarithmic factor comes from the second moment of the sum-of-two-squares representation function [2606.07735].

## 4. Explicit families and representative quartets

A modern explicit polynomial parametrization is given by Choudhry’s degree-three theorem. With four arbitrary parameters \(p,q,r,s\),
\[
x_1=\phi(p,q,r,s),\quad x_2=\phi(p,r,s,q),\quad x_3=\phi(p,s,q,r),\quad x_4=\phi(q,r,p,s),
\]
\[
y_1=\phi(p,q,s,r),\quad y_2=\phi(p,r,q,s),\quad y_3=\phi(p,s,r,q),\quad y_4=\phi(q,s,p,r),
\]
where
\[
\phi(a,b,c,d)=a^2bc+abc^2+ac^2d+acd^2+b^2cd+bc^2d.
\]
This yields an ideal degree-three solution, and the common sums \(\sigma_1,\sigma_2,\sigma_3\) are symmetric functions of \(p,q,r,s\) [2106.13944].

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

| Type | Quartets | Source |
|---|---|---|
| Parametric family | \((x_i),(y_i)\) from \(\phi(p,q,r,s)\) | [2106.13944] |
| Symmetric example | \(\{0,4,7,11\}=_3\{1,2,9,10\}\) | [2603.12320] |
| Example with repetition | \(\{-3,0,1,4\}=_3\{-2,-2,3,3\}\) | [2603.12320] |
| Multigrade chain | \([0,28,29,57]^k=[1,21,36,56]^k=[2,18,39,55]^k=[6,11,46,51]^k,\ (k=1,2,3)\) | [2506.11429] |

The pair \(\{0,4,7,11\}\) and \(\{1,2,9,10\}\) is symmetric about \(c=\tfrac{11}{2}\); after subtracting \(11/2\) and doubling, it becomes
\[
2(A-c)=\{\pm 11,\pm 3\},\qquad 2(B-c)=\{\pm 9,\pm 7\},
\]
and
\[
11^2+3^2=9^2+7^2=130
\]
[2606.07735]. The example \(\{-3,0,1,4\}=_3\{-2,-2,3,3\}\) is singled out in the physics literature as the solution minimizing \(\sum q_i^2\) among the listed charge assignments [2603.12320].

The 2025 survey also records a trigonometric symmetric identity and several huge integer ideal quartets, including both symmetric and non-symmetric examples [2506.11429].

## 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 \(3\) is
\[
\{0,3,5,6,9,10,12,15\},\qquad \{1,2,4,7,8,11,13,14\},
\]
which satisfies equality of sums of powers for \(m=0,1,2,3\) but uses \(8\) terms on each side, not the ideal \(4\) [1411.6168]. The same 8-vs-8 cubic partition is the \(M=2,N=1,p=3\) case of the generalized digit-map construction [2509.11269].

Combinatorial Prouhet constructions yield the same scale. The binary generalized Thue–Morse word
\[
ABBABAABBAABABBA
\]
encodes the partition
\[
\{1,4,6,7,10,11,13,16\}\quad\text{vs.}\quad\{2,3,5,8,9,12,14,15\},
\]
and the cited work reports that \(\mathrm{PTE}(16,2,3)\) contains just one word, namely this length-16 Thue–Morse prefix [1304.6756].

The 2018 digit-sum paper shows how cancellations can beat the classical size barrier. For \(b=2\), \(N=3\), \(x=y=1\), the generalized construction produces
\[
\{0,5,7,8\}\cup\{2,3,5,10\},
\]
with equal first and second power sums; cancelling the common value \(5\) yields the smaller partition
\[
\{0,7,8\}\cup\{2,3,10\},
\]
and the paper explicitly remarks that its results are amenable to a computational search, which may discover new, smaller, solutions to the classical problem [1805.10569]. 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 \(4\), if \(q\mid C_4\), then modulo \(q\) the two multisets coincide after reordering, giving obligatory local constraints [2304.11254]. The 2025 survey also normalizes ideal quartets by
\[
\alpha_i=\frac{a_i}{a_4},\qquad \beta_i=\frac{b_i}{a_4},
\]
and records the interlacing pattern
\[
0\le \alpha_1<\beta_1\le \beta_2<\alpha_2\le \alpha_3<\beta_3\le \beta_4<1,
\]
together with
\[
\beta_4\ge \sin^2\!\left(\frac{3\pi}{8}\right)
\]
[2506.11429]. 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 \(k=2\) or \(3\), and omits the cubic formulas because they were already known [1603.00206]. 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 [2307.11330].

Beyond \(\mathbb Z\), the degree-three ideal problem changes character. Over the Gaussian integers,
\[
\{0,0,0,0\}=_3\{1,-1,i,-i\},
\]
and the associated quartic difference is
\[
z^4-(z^4-1)=1,
\]
so the constant is a unit [1011.1262]. 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
\[
\sum a_i=\sum b_i,\qquad \sum a_i^2=\sum b_i^2,\qquad \sum a_i^3=\sum b_i^3,
\]
so the lower bound \(n\ge k+1\) implies that at least four states are required in the minimal case [2603.12320]. In weighted-path spectral theory, the existence of a \(7\)-chain with periodic cospectral vertices at positions \(0\) and \(4\) is equivalent to a solution of \(PTE_4^0\), i.e. a disjoint ideal degree-three solution [2509.09948]. 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 \(t=o(2^k)\) is identified as the main obstruction to extending those hardness results [1611.03069].

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 [1603.00206], 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 [2606.07735].

Source: https://www.emergentmind.com/topics/ideal-degree-three-prouhet-tarry-escott-problem