Small Sets of k-th Powers

1994 ◽  
Vol 37 (2) ◽  
pp. 168-173
Author(s):  
Ping Ding ◽  
A. R. Freedman
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Nonnegative Integer ◽  
Large Integer

AbstractLet k ≥ 2 and q = g(k) — G(k), where g(k) is the smallest possible value of r such that every natural number is the sum of at most r k-th powers and G(k) is the minimal value of r such that every sufficiently large integer is the sum of r k-th powers. For each positive integer r ≥ q, let Then for every ε > 0 and N ≥ N(r, ε), we construct a set A of k-th powers such that |A| ≤ (r(2 + ε)r + l)N1/(k+r) and every nonnegative integer n ≤ N is the sum of k-th powers in A. Some related results are also obtained.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution

2019 ◽  
Vol 11 (02) ◽  
pp. 1950015
Author(s):  
Rafał Kapelko
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Natural Number ◽  
Uniform Distribution ◽  
Gamma Function ◽  
Order Statistic ◽  
Unit Interval ◽  
Mobile Sensors ◽  
Mean Deviation ◽  
Current Location

Assume that [Formula: see text] mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors [Formula: see text] from [Formula: see text] to [Formula: see text] where the contribution of the [Formula: see text] sensor is its displacement from the current location to the anchor equidistant point [Formula: see text] raised to the [Formula: see text] power, when [Formula: see text] is an odd natural number. As a consequence, we derive the following asymptotic identity. Fix [Formula: see text] positive integer. Let [Formula: see text] denote the [Formula: see text] order statistic from a random sample of size [Formula: see text] from the Uniform[Formula: see text] population. Then [Formula: see text] where [Formula: see text] is the Gamma function.

The þ-function in λ-K-conversion

1937 ◽  
Vol 2 (4) ◽  
pp. 164-164 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Definition Of ◽  
Image Position

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

A NOTE ON ERDŐS–STRAUS AND ERDŐS–GRAHAM DIVISIBILITY PROBLEMS (WITH AN APPENDIX BY ANDRZEJ SCHINZEL)

2013 ◽  
Vol 09 (03) ◽  
pp. 583-599 ◽  
Author(s):  
MACIEJ ULAS ◽  
ANDRZEJ SCHINZEL
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Positive Answer ◽  
Computational Results

In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ+ be fixed. Does there exist a positive integer k such that [Formula: see text] The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that [Formula: see text] The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems.

Paradox of the class of all grounded classes

1953 ◽  
Vol 18 (2) ◽  
pp. 114-114 ◽  
Author(s):  
Shen Yuting
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Image Position ◽  
Special Case

A class A for which there is an infinite progression of classes A1, A2, … (not necessarily all distinct) such thatis said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes.Let us assume that K is a groundless class. Then there is an infinite progression of classes A1, A2, … such thatSince A1 ϵ K, A1 is a grounded class; sinceA1 is also a groundless class. But this is impossible.Therefore K is a grounded class. Hence K ϵ K, and we haveTherefore K is also a groundless class.This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n-circular (n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves (n = 1).More exactly, a class A1 is circular if there exists some positive integer n and classes A2, A3, …, An such thatFor any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, such thatQuite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n-circular, for each positive integer n.

APPROXIMATING GREEN'S FUNCTIONS ON ℙ1 POSITIVE CHARACTERISTIC

2010 ◽  
Vol 12 (04) ◽  
pp. 537-567
Author(s):  
DAESHIK PARK
Keyword(s):  
Positive Integer ◽  
Disjoint Union ◽  
Positive Characteristic ◽  
Green’S Functions ◽  
Finite Union ◽  
Large Integer ◽  
Probability Vector ◽  
Symmetric Set ◽  
Union Of Balls ◽  
Vector Formula

Fix a finite K-symmetric set [Formula: see text] and a K-symmetric probability vector [Formula: see text]. Let 𝔇v be a finite union of balls [Formula: see text] for some ah ∈ Kv and some [Formula: see text], where the balls 𝔅(ah, rh) are disjoint from 𝔛. Put 𝔈v := 𝔇v ∩ ℙ1(Kv). Then there exists a positive integer Nv such that for each sufficiently large integer N divisible by Nv, there are a number Rv, with [Formula: see text], and an [Formula: see text]-function fv(z) ∈ Kv(z) of degree N whose zeros form a "well-distributed" sequence in 𝔈v such that [Formula: see text] is a disjoint union of balls centered at the zeros of fv(z) and for all z ∉ 𝔇v, [Formula: see text]

scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 339 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Nonnegative Integer ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

scholarly journals ON THE WARING–GOLDBACH PROBLEM FOR CUBES

2009 ◽  
Vol 51 (3) ◽  
pp. 703-712 ◽  
Author(s):  
JÖRG BRÜDERN ◽  
KOICHI KAWADA
Keyword(s):  
Natural Number ◽  
Large Integer ◽  
Natural Numbers ◽  
Goldbach Problem ◽  
Almost All

AbstractWe prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.

Sign in / Sign up

Export Citation Format

Share Document