Tarry-Escott Problem Research Articles

In Memoriam: Peter Benjamin Borwein (1953–2020)

Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.

Newton polygons and the constant associated with the Prouhet–Tarry–Escott problem

Newton polygons and the constant associated with the Prouhet–Tarry–Escott problem

Ideal solutions in the Prouhet–Tarry–Escott problem

For given positive integers m m and n n with m > n m>n , the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size n n having identical k k th moments for 1 ≤ k ≤ m 1\leq k\leq m ; in the ideal case one requires m = n − 1 m=n-1 , which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Z \mathbb {Z} and over the ring of integers of several imaginary quadratic number fields. Over Z \mathbb {Z} , we significantly extend searches for symmetric ideal solutions at sizes 9 9 , 10 10 , 11 11 , and 12 12 , and we conduct extensive searches for the first time at larger sizes up to 16 16 . For the quadratic number field case, we find new ideal solutions of sizes 10 10 and 12 12 in the Gaussian integers, of size 9 9 in Z [ i 2 ] \mathbb {Z}[i\sqrt {2}] , and of sizes 9 9 and 12 12 in the Eisenstein integers.

Limitations of Mean-Based Algorithms for Trace Reconstruction at Small Edit Distance

Trace reconstruction considers the task of recovering an unknown string <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {x}\in \{0,1\}^{n}$ </tex-math></inline-formula> given a number of independent “traces”, i.e., subsequences of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {x}$ </tex-math></inline-formula> obtained by randomly and independently deleting every symbol of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {x}$ </tex-math></inline-formula> with some probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> . The information-theoretic limit of the number of traces needed to recover a string of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is still unknown. This limit is essentially the same as the number of traces needed to determine, given strings <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {x}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {y}$ </tex-math></inline-formula> and traces of one of them, which string is the source. The most-studied class of algorithms for the worst-case version of the problem are “mean-based” algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show that, on the one hand, distinguishing strings that are nearby in Hamming distance is “easy” for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is “hard” for mean-based algorithms. Along the way, we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Furthermore, we show that the converse is also true, thus, finding explicit solutions to the PTE problem is equivalent to the problem of finding explicit strings that are hard-to-distinguish by mean-based algorithms. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes’ rule of signs for polynomials over the reals.

The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations

In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A subseteq {1, dots , n} in k-sets such that the first k-1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial f_A(x) := prod _{x in A} (x-a). Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation f_A(x)=P(y) where P(y)in mathbb {Q}[y] and on shifted power values of f_A(x).

Arity Shape of Polyadic Algebraic Structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.

Settling some sum suppositions

We solve multiple conjectures by Byszewski and Ulas about the base b sum-of-digits function. In order to do this, we develop general results about summations over the sum-of-digits function. As a corollary, we describe an unexpected new result about the Prouhet–Tarry–Escott problem. In some cases, this allows us to partition fewer than bN values into b sets $${\{S_1,\ldots,S_b\}}$$ such that $$\sum_{s\in S_1}s^k = \sum_{s\in S_2}s^k = \cdots = \sum_{s\in S_b}s^k $$ for $${0\leq k \leq N-1}$$ . The classical construction can only partition bN values such that the first N powers agree. Our results are amenable to a computational search, which may discover new, smaller solutions to this classical problem.

A new approach to the Tarry–Escott problem

In this paper we describe a new method of obtaining ideal solutions of the well-known Tarry–Escott problem, that is, the problem of finding two distinct sets of integers [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is a given positive integer. When [Formula: see text], only a limited number of parametric/numerical ideal solutions of the Tarry–Escott problem are known. In this paper, by applying the new method mentioned above, we find several new parametric ideal solutions of the problem when [Formula: see text]. The ideal solutions obtained by this new approach are more general and, very frequently, simpler than the ideal solutions obtained by the earlier methods. We also obtain new parametric solutions of certain diophantine systems that are closely related to the Tarry–Escott problem. These solutions are also more general and simpler than the solutions of diophantine systems published earlier.

Newton polygons and the Prouhet–Tarry–Escott problem

We show how Newton polygons can be used to obtain new information on the 2-adic valuations of a certain constant associated with the Prouhet–Tarry–Escott (PTE) problem. Two explicit examples are given involving the cases n=8 and n=9 in the PTE problem.

On trees with the same restricted [formula omitted]-polynomial and the Prouhet–Tarry–Escott problem

This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism.

Solutions to the multi-dimensional Prouhet–Tarry–Escott problem resulting from composition of balanced morphisms

We show that the structure of a multi-dimensional word obtained by composition of a finite sequence of morphism-like mappings, k of them being balanced, induces a solution to the multi-dimensional Prouhet–Tarry–Escott problem of degree k.

On the number of solutions of the Tarry-Escott problem of degree two and the related problem over some finite fields

The number of ideal solutions over $${\mathbb Z}_p$$ of the Tarry-Escott problem of degree two and the number of solutions over $${\mathbb Z}_p$$ of the related system of equations: $$\sum _{i=1}^{3} a_i^r = \sum _{i=1}^{3} b_i^r \quad \text {for}\quad r = 1,2,5$$ are completely determined.

On Prouhet’s solution to the equal powers problem

Eugène Prouhet is considered to be one of the fathers of combinatorics on words. In his pioneering work, he constructed a solution to the equal powers problem in number theory (later known as the Prouhet–Tarry–Escott problem) based on positions of symbols in words obtained by iteration of a uniform endomorphism of the free monoid over a finite alphabet. We show that composition of uniform morphisms of free monoids is a fairly powerful tool to obtain solutions to a more general problem, which covers the major generalizations of Prouhet’s theorem appearing in literature.

The Prouhet-Tarry-Escott problem for Gaussian integers

Given natural numbers $n$ and $k$, with $n>k$, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of $\Z$, say $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_n\}$, such that \[x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for $i=1,...,k$. Many partial solutions to this problem were found in the late 19th century and early 20th century. When $n=k-1$, we call a solution $X=_{n-1}Y$ ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.

The Tarry-Escott problem of degree two

The optimum reduced solutions of degree two of the Tarry-Escott problem over a ring of Gaussian integers and over a ring of polynomials over a finite field are completely determined.

Some diophantine problems concerning equal sums of integers and their cubes

This paper gives a complete four-parameter solution of the simultaneous diophantine equations $x+y+z=u+v+w, x^3+y^3+z^3=u^3+v^3+w^3,$ in terms of quadratic polynomials in which each parameter occurs only in the first degree. This solution is much simpler than the complete solutions of these equations published earlier. This simple solution is used to obtain solutions of several related diophantine problems. For instance, the paper gives a parametric solution of the arbitrarily long simultaneous diophantine chains of the type $x^k_1+y^k_1+z^k_1=x^k_2+y^k_2+z^k_2=\ldots=x^k_n+y^k_n+z^k_n=\ldots,~~k=1,3.$ Further, the complete ideal symmetric solution of the Tarry-Escott problem of degree $4$ is obtained, and it is also shown that any arbitrarily given integer can be expressed as the sum of four distinct nonzero integers such that the sum of the cubes of these four integers is equal to the cube of the given integer.

Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.

This paper is concerned with the system of simultaneous diophantine equations $\sum_{i=1}^6A_i^k=\sum_{i=1}^6B_i^k$ for $k=2, 4, 6, 8, 10.$ Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree $11$, that is, of the system of simultaneous equations, $\sum_{i=1}^{12}a_i^k=\sum_{i=1}^{12}b_i^k$ for $k=1, 2, 3,\ldots,11.$ We use one of the ideal solutions to prove new results on sums of $13^{th}$ powers. In particular, we prove that every integer can be expressed as a sum or difference of at most $27$ thirteenth powers of positive integers.

The two-dimensional Prouhet–Tarry–Escott problem

In this paper we generalize the Prouhet–Tarry–Escott problem (PTE) to any dimension. The one-dimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n , k ∈ N , for two different multi-sets { ( x 1 , y 1 ) , … , ( x n , y n ) } , { ( x 1 ′ , y 1 ′ ) , … , ( x n ′ , y n ′ ) } of points from Z 2 such that ∑ i = 1 n x i j y i d − j = ∑ i = 1 n x i ′ j y i ′ d − j for all d , j ∈ { 0 , … , k } with j ⩽ d . We present parametric solutions for n ∈ { 2 , 3 , 4 , 6 } with optimal size, i.e., with k = n − 1 . We show that these solutions come from convex 2 n-gons with all vertices in Z 2 such that every line parallel to a side contains an even number of vertices and prove that such convex 2 n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers.

Minimal set of class-sums characterizing the ordinary irreducible representations of the symmetric group, and the Tarry-Escott problem

The shape of a Young diagram Y (| Y| = n) can be specified in terms of the set of symmetric power sums over its contents, σ l = Σ ( ij) ϵY ( j − i) l ; l = 1, 2, …, n. It is remarkable that the set of power sums σ 1, σ 2, …, σ k is sufficient to characterize the Young diagrams possessing up to n( k) boxes, where n( k) is considerably larger than k. Numerical evidence for k⩽5 is roughly consistent with n(k) ∼ 4( 4 3 ) 2k . The lower bound n( k) > k + max(2, √ k) has been derived by examination of some properties of Young diagrams with a large number of rows, and the upper bound n( k) < 2 2 k+1 has been established using a variant of the Tarry-Escott problem.

On the Construction of Sets of Integers with Equal Power Sums

In this paper we give a new method for constructing sets of integers having equal t th power sums. Using the method, some new results are derived concerning the Tarry-Escott Problem.