2. Preliminaries
In this section, the authors recall some basic definitions and results that play a significant role in the study of the PTE problem.
Definition 1. The Prouhet Tarry Escott problem aims to obtain two distinct sets of integers say and such that:where n is known as the size and k is known as the degree. The solution sets and are usually represented as . Example 1. The two sets and satisfy the conditions:Thus, these two sets serve as a solution of Prouhet Tarry Escott problem of degree 3 and size 8. If the size and the degree of the PTE problem differ by one (i.e., degree = size − 1), then the solutions are known as ideal solutions, otherwise they are called non-ideal solutions. For example, the two distinct sets of integers and constitute an ideal solution of the PTE problem of size 2 and degree 1 whereas the sets and exemplify a non-ideal solution of the PTE problem of degree 2 and size 4.
Note 1. A solution of the PTE problem in which merely form a permutation of is called trivial.
Definition 2. Consider the PTE problem of odd size n. Suppose the set of integers, say satisfies the system of equations , then . In this case, the solution is called an odd ideal symmetric solution. If the size of the PTE problem is even and suppose where the are positive integers, then:and in this case the solution becomes an even ideal symmetric solution. Example 2. The sets and provide an ideal symmetric solution, whereas and give an ideal non-symmetric solution for the PTE problem of and , respectively.
One can also define the constant associated with the PTE problem as follows:
Definition 3. Suppose that and are two sets of positive integers satisfying with . Then statements (1) and (2) are equivalent: This implies that and is a constant. This constant is known as the constant associated with the PTE problem.
It is to be noted that the equations in Definition 3 are derived from the concept of polynomial equivalence (Newton’s equations).
General Theorems
One of the fundamental results on solutions of the PTE problem is due to Frolov [
4].
Theorem 1. If , then where M and K are arbitrary integers.
If one generates a new solution from an existing solution by the repeated application of Theorem 1, then these two sets of solutions are called equivalent solutions. Solutions which are not equivalent are considered while discussing distinct solutions. In 1910, Escott [
5] established Theorem 2.
Theorem 2. If and are two polynomials with integral coefficients having their first r terms alike, then and where d is an integer, are two polynomials having their first terms alike.
Later Tarry [
6] developed a theorem similar to Theorem 2.
Theorem 3. If is a solution set of the PTE problem of degree k, then where h is an arbitrary integer, is a solution of the PTE problem of degree .
Another important result on the PTE problem is due to Bastien [
7] which is stated as follows.
Theorem 4. If the system of equations , has a non-trivial solution, then .
Theorem 5 ([
8])
. If , then where T is an arbitrary integer. Theorem 6. Let be two multisets of integers that constitute an ideal PTE solution, and suppose that a prime p divides the constant associated with this solution. Then we can reorder the integers so that for .
Let and suppose satisfy . Any set such that is denoted as an set. If , every set is an set and conversely. If , we say that , the sum of over all sets and is independent of r if .
Now, let us recall the well known Lehmer’s result on equal sums of like powers.
Theorem 7. If are any numbers and , then is independent of r for .
Another interesting idea used in the study of PTE problems is the concept of Prouhet–Thue–Morse sequence.
Definition 4. The Prouhet–Thue–Morse sequence is a binary sequence starting with 0 and the nth element is defined as if the number of ones in the binary expansion of n is odd, and if the number of ones in the binary expansion of n is even.
Theorem 8, originally given by Prouhet, illustrates the relation between the Prouhet–Thue–Morse sequence and the PTE problem.
Theorem 8 ([
9])
. The Prouhet–Thue–Morse sequence has the following property. Define , . Then for , we have . Remark 1. Two important facts to be considered in the study of the PTE problem are as follows:
- 1.
No integer can appear more than twice among either or .
- 2.
It is not possible to have three consecutive integers on one side of the solution or two repeated integers and a consecutive integer.
3. On the Prouhet Tarry Escott Problem
There have been a lot of studies performed on the solutions of the PTE problem worldwide. It was Euler [
10] who made the first contribution to the PTE problem. He remarked that
where
are positive integers. Later, Goldbach [
11] pointed out in a letter to Euler that if
and
s are positive integers, then:
and the findings of Euler was a special case,
, of what Goldbach noticed. Pollock [
12] and Proth [
12] also provided some solutions to this problem as a continuation of Goldbach’s findings. Cesaro [
12] demonstrated that the integers
satisfies the relation:
and showed some trigonometrical properties of these three sets of four integers.
Later, Prouhet [
12] noted that the first 27 integers;
could be separated into three sets out of which two sets say
and
satisfy
as well as
. Prouhet generalized this property of integers as: “The
positive integers can be separated into
n sets consisting of
terms so that the sum of the
kth powers; (
) of terms is the same for all these sets”. Even though Prouhet contributed a first solution to the PTE problem, the problem was known as Tarry Escott problem or equal size power’s problem until the 20th century when Prouhet’s solution was rediscovered. Now the problem is generally termed as the Prouhet Tarry Escott problem. Similar kinds of earlier studies on the PTE problem can be found in Reference [
12], which contains a good collection of earlier literature.
Several works [
13,
14,
15,
16,
17,
18,
19,
20] discussed both ideal and non-ideal solutions of the PTE problem by adopting different elementary approaches. Chernick [
21] obtained ideal solutions of the PTE problem for the cases
and 7 including the solutions that Escott [
12] had produced for the cases
and 5. Chernick also obtained all rational solutions for the case
together with the parametric solutions. Burchnall and Chaundy [
22] explained how the solutions of the PTE problem could be represented in the form of magic squares. Later, Dorwart and Brown [
23] presented some general theorems regarding the existence of solutions of the PTE problem in a more descriptive way. They aimed to give detailed examples instead of proving those theorems and explained how to obtain equivalent solutions once an initial solution was obtained. Dorwart and Brown also proved a beneficial result regarding the least number
, for which the PTE problem
possess non-trivial solutions. They provided some examples to ensure
for
and tabulated the values for
and
. Also, they mentioned how the ideal solutions of the PTE problem could be conveniently applied to compute the large number of decimal places occurring in the logarithm of integers. They considered a familiar series
. Then they replaced
M and
N by univariate polynomials with integral roots such that
is a constant. They also pointed out a connection between the PTE problem and the derivation of the following formula, used for the computation of
,
, where
X is a real polynomial whose degree and the number of fractions on the left-hand side of the equality are equal. They observed that the problem of reducibility of polynomials of the form
, where
p is a prime and
, is equivalent to finding the ideal solutions of the PTE problem. So, the work in Reference [
23] seems to be an important building block in the area of the PTE problem. As a continuation, Dorwart [
24] invented some equivalent forms of solutions for ideal solutions with
, presented a general non-symmetric solution for
, and provided a base for finding ideal parametric solutions of the PTE problem with
.
Another significant contribution in the early stages was from Wright [
25]. He provided two elementary proofs for Lehmer’s result, while Lehmer’s proof depends on the ideas of generating functions, exponential differentiation, matrices, and complex roots of unity. Wright provided two proofs. One depends on the factor theorem for polynomials, and the other depends on the multinomial theorem for a positive integral index. Wright also provided a tool for extracting solutions of the PTE problem with arbitrary degree
k. He proved that if a non-trivial solution of
, where
and
, exists for
and
, then we can construct a non-trivial solution for the same
and
, where
m is any positive whole number. He also constructed a solution for general
s with
provided
. Later, Roberts [
26] presented a new proof of Lehmer’s theorem by using difference operators and provided some solutions of the PTE problem as a particular case. The proof given in Reference [
26] was more general than the already existing results.
Additionally, Wright [
27] provided a direct proof for Prouhet’s generalization. One of the major results in Reference [
27] was as follows: “Express each
as a decimal in the scale of
j. If the least positive residue to modulus
j of the sum of the digits of
n in this scale is
, then assign
n to the set
. Then each of the sets
contains just
members, which may be taken as
and which satisfy the simultaneous equations
, with
where
,
, and
.” After two years, Melzak [
28] implemented a method for estimating the bound
, the least integer in the set of all
s (size of the problem) that possesses a solution for the PTE problem, which yields to the actual solution. He proved that “
, where
is the class of all polynomials having integer coefficients,
N is a positive integer, and
for
”. This gives an exact expression for finding an estimate for
rather than computing the value of
. The proof given in Reference [
28] was non-constructive, and also could not be employed to compute
, however it led to better-bound estimates of
than those previously existing. He also provided a list of estimates for
for different
n. Later, Barrodale [
29] provided some new solutions to the problem with the help of computers, by improving the lower bounds that had been found by Melzak. He extended the results to
. He also discussed some extension of the PTE problem by defining
as the least value of
s such that
where
, has a solution together with
. Sinha [
30] obtained a parametric solution of size 10 and degree 8 by solving the system
where the
’s and
’s are integers. Eggleton and Selfridge [
31] created a set of consecutive integers such that the integers do not have comparatively big prime factors based on the PTE problem solutions.
Adler and Robert Li [
32] implemented the method of construction of magic cubes of some order
N from the generalized Morse–Hedlund sequence, which is a particular case of the Prouhet sequence. Also, they clearly explained how solutions of the PTE problem are associated with magic cubes and concluded by proving a nice result regarding the entries in the magic cubes. According to the result in Reference [
32], if a magic cube of order
is considered, then its conjugate sub-cubes having dimension
n satisfy the equal sum of the
kth power of their inputs, where
. Later, Bremner [
33] geometrically developed the problem of equal sums of fifth powers by studying a K3 surface contained in the fourfold. Compared to other approaches, it was a new one in the study of equal sums of like powers. So it opened a new technique that could be shortly applied in the study of the PTE problem. Thus, the contribution from this work is highly important in the study of the PTE problem. Nine years later, Rees and Smyth [
34] studied the constant associated with the PTE problem, which was a new approach in the study of the PTE problem at that time. They proved some divisibility results on the constant
associated with the PTE problem for
. They also studied the divisibility of
modulo a prime power.
After a year, Smyth [
35] again studied ideal 9th order multigrades and Letac’s elliptic curve. He provided a constructive definition of 9th order multigrades as follows: “9th order multigrades are the solutions
in integers of the system of equations
for
”. Later, Borwein and Ingalls [
36] presented some new limits for the Easier–Waring problem, irrespective of the existing bounds and also for the PTE problem of small size. They also presented an algorithm for finding ideal symmetric
p-adic solutions for the PTE problem and with the help of those algorithms, they started to find an ideal symmetric solution of size 11. Jacroux [
37] derived some new results on the construction of sets of integers with equal power sums. He explained the constructive method as follows: “Let
and
be sets of integers with
such that
and
. Then
where for
and
is the set
with
added to each element”. Jacroux also implemented a new idea in partitioning the set of integers, which generalizes Prouhet’s following findings: “Let
. Then the first
positive integers can be partitioned into
L classes
of size
d such that
if and only if
L is odd or
d is even.” Ekl [
38] employed a computational technique to find the solution of
. Maltby [
39] studied the relation between pure product polynomials and the PTE problem. In the theory of pure product polynomials, the PTE problem is to find multigrades of the smallest possible size for each degree.
A lot of work in different directions on the PTE problem has been done from 2000 onwards. Choudhry [
40] introduced the complete ideal symmetric solutions of the 4th-degree PTE problem. He discussed the non-symmetric solution for
and used it to obtain a parametric solution of the corresponding Diophantine system
for
and 6. Later, Borwein [
41] provided a fundamental tool in computational techniques that could be applied to the PTE problem. Also, he presented a list of the smallest known ideal solutions of size 2 to 12, excluding the case of size 11.
Figure 1 illustrates these solutions.
Later, Borwein et al. [
42] explained a computational technique for finding the ideal symmetric solutions to the PTE problem. They considered Theorem 6 in order to reduce the search limit. They started with Newton’s iteration and a value above the search limit, and continued until either the iteration converges, the iteration fails to monotonically decrease, or the 60th iteration was reached. They obtained solutions of sizes up to 12 and presented two new solutions of size 10.
Choudhry [
43] extended previous analysis and developed ideal non-symmetric solutions of the PTE problem of degrees four and five. He also discussed the parametric solutions of the Diophantine system related to these problems. He obtained a much simpler parametric solution of degree four than in Reference [
40]. Again, Choudhry [
44] studied the solutions of the Diophantine system connected to the PTE problem with no non-trivial solutions. Later, Choudhry [
45] introduced the matrix analogs of the PTE problem by considering the problem over
matrices over
,
. Srikanth and Gopalan [
46] obtained a general form of non-ideal, non-trivial, parametric integral solutions of the PTE problem with degree
and size
.
In 2007, Alpers and Tijdeman [
47] studied the two-dimensional Tarry–Escott problem. They presented parametric solutions for the cases
and proved the Prouhet theorem for the two-dimensional case. Also, they proved that the solutions of the two-dimensional problem give rise to solutions of the classical PTE problem. They discussed the geometrical aspects of the PTE problem by obtaining solutions from convex
-gons whose vertices belong to
and also introduced the study of the classical PTE problem over Gaussian integers.
Choudhry and Wróblewski [
48] introduced an idea to obtain solutions to the PTE problem of degree 11, from the solutions of the system
for
. Also, they proved new results on the sum of thirteenth powers with the help of those ideal solutions. At the end of their research article, they showed that any integer could be written as an addition or subtraction of 13th powers of at most 27 positive integers. Prugsapitak [
49] studied the PTE problem over a quadratic field. Again, Choudhry [
50] produced the complete ideal symmetric solution of the 4th-degree PTE problem and proved that any integer
n could be represented as
where
are distinct non-zero integers satisfying
.
Two years later, Prugsapitak [
51] determined the complete ideal solutions of the PTE problem of degree two over both
, the ring of Gaussian integers, and
, the ring of polynomials over a finite field
where
p is a prime. A different approach to that of Cayley [
52,
53] was adopted. Also, a discussion on obtaining ideal solutions over
from the integer ideal solutions of degree two and from the proper integer solutions of the two-dimensional PTE problem was performed. The developed method could be used to get infinitely many solutions of degree two over
. Cayley [
52,
53] implemented solutions of the classical PTE problem over Gaussian integers. He performed both the theoretical and computational techniques to analyze the ideal solutions over
and provided some divisibility results on the constant associated with the PTE problem over
and
.
Table 1 provides the list of primes dividing the constant
associated with the ideal symmetric solutions of the classical PTE problem.
Černý [
54] presented a class of solutions to the PTE problem. Prouhet’s solution was described as a special case of that class. Again, Choudhry [
55] provided a detailed study of the PTE problem by considering a new condition
together with the already existing conditions of the PTE problem. He provided numerical solutions for many Diophantine systems of this type. Additionally, he showed that certain Diophantine systems with equal sums of powers and equal products could not have any non-trivial solutions.
Later, Kongsiriwong and Prugsapitak [
56] investigated the number of ideal solutions to the second-degree PTE problem over the finite field
, where
p is a prime. They presented the non-trivial integer solutions of the corresponding Diophantine system
, for
. They first established the result over
, where
is an odd prime, using Newton’s identities that relate the two types of symmetric polynomials. Choudhry [
57] extended his previous studies to obtain the minimum number of terms required for the Diophantine system
, where
, associated with the PTE problem
, where
.
Nguyen [
58] showed that the solution of the PTE problem can be obtained by deriving the generalization of the product generating function for the Prouhet–Thue–Morse sequence. Also, he presented some applications of Prouhet’s solutions in the radar field and he provided an elementary and straightforward proof for results regarding the generalization of the product generating function. Černý [
59] again proved that the structure of a multi-dimensional word obtained by a composition of a finite sequence of morphism-like mappings, out of which
k were balanced, yields a solution to the multi-dimensional PTE problem of degree
k. Also, he revealed the solutions to the multi-dimensional PTE problem over an arbitrary semi-ring.
The application of the gravitational search algorithm (GSA) to find numerous solutions of the PTE problem was investigated by Venkatesan et al. [
60]. They provided computational results to establish that heuristic methods like GSA give high-quality solutions to the PTE problem. The bridge between non-polynomial (NP)-hardness of the Reed–Solomon decoding and the PTE problem was the subject of Gandikota et al. [
61]. They also discussed the existence of inhomogeneous PTE solutions over an arbitrary field. Laughlin and Zimmer [
62] provided a relation between the well poised (WP)-Bailey pair and the PTE problem. They also produced a simple example to illustrate the relationship. They used the generalized telescoping approach to prove their results on the PTE problem. Also, they noted that the ideal solutions of the PTE problem yield an infinite summation formula, with a suitable example. Recently, Filaseta and Markovich [
63] studied the 2-adic values of the constant associated with the PTE problem. They produced the 2-adic values for the cases
and
using the classical theory of the Newton polygon. For the case
, they found a lower bound; whereas for the case
, they succeeded to produce the exact value. Again, Choudhry [
64] found out some new parametric ideal solutions of the problem with degree less than or equal to 7, by applying the concept of arithmetic progressions. He invented some new parametric solutions of the Diophantine systems associated with the PTE problem, which he considered as a specimen for the research. Also, he suggested that the method can be extended to the cases when
and
, and he left it as an open problem. Shuwen [
65] provided a list of ideal symmetric solutions of the PTE problem for
and ideal non-symmetric solutions for
together with some elementary theorems.
Thus, a detailed survey has been performed related to the PTE problem which summarizes the research space. All of this work collectively provides a good picture of the development of significant outcomes in this area.
4. Applications
The ideal solutions of the PTE problem play a major role in fields like combinatorics, computer science, the easier waring problems, andin finding the rational points on elliptic curves. There are some applications of solutions of the PTE problem on trace reconstruction problems. Krasikov and Roditty [
66] established the k-reconstructibility of words of length
n for
by dealing with words only over the alphabet
. They also proved the necessary condition for non-reconstructibility of words of length
n. Later, Borwein et al. [
67] proved that the classical PTE problem could be rephrased as the problem of finding the maximal vanishing at 1 of a polynomial with integer coefficients and with
norm
. Later, Foster and Krasikov [
68] dealt with the problem of finding, given
n, the smallest
m such that there exists a polynomial
of degree
m satisfying
and showed that
. Also, they improved the value of
m for polynomials non-negative on
, compared to the result in Reference [
67], and they sharpened the bounds concerning the PTE problem and the sequence reconstruction problem.
Another important application of the PTE problem is its relation with tomography. The first published result on the relation between a particular PTE problem and tomography can be seen in Alpers’ Ph.D. thesis [
69]. Also, he showed that the switching components yield PTE solutions for specific values of
. Hernández and Luca [
70] showed how graph theory relates to the ideal solutions of the PTE problem. They proved the necessary and sufficient condition for the existence of integer roots of chromatic polynomials of a non-chordal graph with ideal solutions of the PTE problem. Also they proved that, if a family
of graphs whose chromatic polynomials have integer roots is considered, then
if and only if the degree of the ideal solution of the PTE problem is
, for
. Here,
stands for the graph made from the disjoint union of
k complete graphs on
vertices. Later, Broadhurst [
71] found ten consecutive factorizations by using the solutions of the ideal PTE problem. He also explained the implementation of the Chinese remainder theorem in the study of the ideal PTE problem by producing some examples, and obtained a new solution to degree 12, with the help of computer programs.
Another interesting application of the PTE problem was provided by Alpers and Tijderman [
47]. They established a direct connection between tomography and a generalization of the PTE problem. They introduced a more general PTE
problem as follows: “Given natural numbers
, find two different multi sets
,
both contained in
and
,
for
such that
, for all non-negative integers
with
.” Another major work was from McLaughlin [
72]. He showed how an amazing identity of Ramanujan was related to the ideal solutions of the PTE problem of size six. He proved that if a sequence of integers
, and
are defined by some generating functions say,
,
,
,
,
,
,
,
,
,
, and
, then for
and each
,
From the example given in Reference [
72], one could easily recognize the relation between the Ramanujan identity and the PTE problem.
A nice application of the PTE problem in the field of computer science was clearly depicted by Borchert et al. [
73]. They studied complete factorization of polynomials of degree 4, 8, and 16 into distinct roots with the help of electronic circuits. Also they proved how one could generate new solutions for the PTE problem using circuits and product gates. Alpers and Larman [
74] derived some bounds on the sizes of solutions to both the classical PTE problem and the two-dimensional PTE problem. They also reported that polynomial size solutions for the two-dimensional PTE problem exist. The result developed by them, that sharpens the bound on the size of solutions is as follows: “For every
, there exists a constant
such that there are solutions of the two-dimensional PTE problem of degree
k and size bounded by
”. Additionally, they remarked at the end of their article that solutions of the classical PTE problem could be obtained by applying a suitable linear functional to two-dimensional PTE solutions that maps
to
, where
are suitably chosen.
One of the applications of PTE solutions in combinatorics is the formation of the generalized Thue–Morse sequence to finite alphabets using PTE partitions of arbitrarily high symmetry. Bolker et al. [
75] showed how such partitions could be used to pour the same volume of coffee from a container into a finite number of cups so that each gets almost the same amount of caffeine. The basic idea used in Reference [
75] was as follows: “To fill two cups with four pours use the word ABBA: pour the first and last quarters into cup A and the second and third quarters into B. For eight pours the magic word is ABBA BAAB. Continue recursively by appending to each sequence of length
n its complement (in the obvious sense) in order to find the optimal partition for pourings using
subdivisions. Collecting all the solutions into an infinite word AB BA BAAB BAABABBA⋯ produces the Thue–Morse sequence. Richman’s argument, showing that the word ABBA BAAB solves the two cup problem using eights, depends essentially on
and
. These equations say that the partition
, whose blocks are the positions of A and B in the magic word, solves an instance of the PTE problem.” With this idea, Bolker et al. generalized the recursive construction of the Thue–Morse sequence in order to generate a new family of solutions to the PTE problem.
Recently, Alpers [
76] discussed the PTE problem in a 2018 habilitation thesis, which contains another recent review of the PTE problem. It included the different tomographic applications in aspects of Diophantine number theory, tomographic super-resolution imaging, tomographic point tracking, tomographic reconstructions of polycrystals, and geometric methods for electron tomography.