scholarly journals A construction for sets of integers with distinct subset sums

10.37236/1341 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Tom Bohman
Keyword(s):  
Upper Bound ◽  
Distinct Subset ◽  
Subset Sums ◽  
Positive Integers

A set S of positive integers has distinct subset sums if there are $2^{|S|}$ distinct elements of the set $\left\{ \sum_{x \in X} x: X \subset S \right\} . $ Let $$f(n) = \min\{ \max S: |S|=n {\rm \hskip2mm and \hskip2mm} S {\rm \hskip2mm has \hskip2mm distinct \hskip2mm subset \hskip2mm sums}\}.$$ Erdős conjectured $ f(n) \ge c2^{n}$ for some constant c. We give a construction that yields $f(n) < 0.22002 \cdot 2^{n}$ for n sufficiently large. This now stands as the best known upper bound on $ f(n).$

scholarly journals ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

2021 ◽  
pp. 1-7
Author(s):  
SÁNDOR Z. KISS ◽  
VINH HUNG NGUYEN
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Distinct Subset ◽  
Subset Sums ◽  
Asymptotic Basis ◽  
Positive Integers

Abstract Let k and l be positive integers satisfying $k \ge 2, l \ge 1$ . A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$ . About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$ ? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals The Minimum Size of Signed Sumsets

10.37236/4881 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Béla Bajnok ◽  
Ryan Matzke
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Minimum Size ◽  
Positive Integers

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\}$$ and$$\rho_{\pm} (G, m, h) = \min \{ |h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.

scholarly journals Integer sets with distinct subset-sums

1988 ◽  
Vol 50 (181) ◽  
pp. 297-297 ◽  
Author(s):  
W. F. Lunnon
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

A Note on the Erdös Distinct Subset Sums Problem

2021 ◽  
Vol 35 (1) ◽  
pp. 322-324
Author(s):  
Quentin Dubroff ◽  
Jacob Fox ◽  
Max Wenqiang Xu
Keyword(s):  
Distinct Subset ◽  
Subset Sums

scholarly journals The number of distinct subset sums of a finite set of vectors

1993 ◽  
Vol 63 (2) ◽  
pp. 234-256
Author(s):  
Ernest Brickell ◽  
Michael Saks
Keyword(s):  
Distinct Subset ◽  
Finite Set ◽  
Subset Sums

scholarly journals A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

2018 ◽  
Vol 8 (1) ◽  
pp. 109-114
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Rational Solution ◽  
Computable Function ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Positive Integers

Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

scholarly journals Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

2016 ◽  
Vol 160 (3) ◽  
pp. 477-494 ◽  
Author(s):  
J. CILLERUELO ◽  
M. Z. GARAEV
Keyword(s):  
Upper Bound ◽  
Absolute Constant ◽  
Primitive Root ◽  
Multiple Roots ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Primitive Root Modulo ◽  
Image Position ◽  
Almost All ◽  
Positive Integers

AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.

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