scholarly journals Monochromatic sequences whose gaps belong to {d, 2d, …, md}

1998 ◽  
Vol 58 (1) ◽  
pp. 93-101 ◽  
Author(s):  
Bruce M. Landman
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Order Of Magnitude ◽  
Positive Integers

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals On a Problem of Erdös and Szekeres

1961 ◽  
Vol 4 (1) ◽  
pp. 7-12 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Lower Bound ◽  
Order Of Magnitude ◽  
Image Position ◽  
Positive Integers

Write12where the maximum is over all real θ, and the lower bound is over all sets of positive integers a1 ≤ a2 ≤ … ≤ an. The problem of the order of magnitude of f(n) was posed by Erdös and Szekeres [1], side by side with a number of other interesting questions. Writing g(n) = log f(n), it is obvious that g(n) is sub-additive, in the sense that g(m+n) ≤ g(m) + g(n), and also that g(1) = log 2, so that g(n) ≤ n log 2.

On the Continued Fraction Expansion of Fixed Period in Finite Fields

2015 ◽  
Vol 58 (4) ◽  
pp. 704-712 ◽  
Author(s):  
Hela Benamar ◽  
Amara Chandoul ◽  
M. Mkaouar
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Finite Fields ◽  
Continued Fraction ◽  
Period Length ◽  
Continued Fraction Expansion ◽  
Positive Integers ◽  
Fixed Period

AbstractThe Chowla conjecture states that if t is any given positive integer, there are infinitely many prime positive integers N such that Per() = t, where Per() is the period length of the continued fraction expansion for . C. Friesen proved that, for any k ∈ ℕ, there are infinitely many square-free integers N, where the continued fraction expansion of has a fixed period. In this paper, we describe all polynomials for which the continued fraction expansion of has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials Q such that deg Q = 2d and Per =t.

scholarly journals Collections of sequences having the Ramsey property only for few colours

1997 ◽  
Vol 55 (1) ◽  
pp. 19-28
Author(s):  
Bruce M. Landman ◽  
Beata Wysocka
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Exact Formula ◽  
Ramsey Property ◽  
Order Of Magnitude ◽  
The Difference ◽  
Positive Integers

A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g(r)(k) such that for every r-colouring of {1, 2, …, g(r)(k)} there is a monochromatic k-term member of 𝑐. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x1, …, xk} such that xi − xi−1 ≡ a (mod m) for i = 2, …, k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let be the collection of sequences that are either a(mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, has the 2-Ramsey property, and that the 2-Ramsey function , corresponding to k-term a(mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for . We show that if a ≠ 0, there exist 4-colourings or 6-colourings (depending on m and a) of the positive integers which avoid 2-term monochromatic members of , but that there never exist such 3-colourings. We also give an exact formula for .

scholarly journals The Cyclic Triangle-Free Process

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 955
Author(s):  
Yu Jiang ◽  
Meilian Liang ◽  
Yanmei Teng ◽  
Xiaodong Xu
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Independent Set ◽  
Ramsey Number ◽  
Free Process ◽  
Vertex Transitive ◽  
Cyclic Graphs ◽  
Positive Integers ◽  
General Graphs ◽  
Independence Numbers

For positive integers s and t, the Ramsey number R ( s , t ) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R ( 3 , t ) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R ( 3 , t ) for small t’s are computed, and R ( 3 , 35 ) ≥ 237 , R ( 3 , 36 ) ≥ 244 , R ( 3 , 37 ) ≥ 255 , R ( 3 , 38 ) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R ( 3 , t ) are proposed.

LOWER BOUNDS ON THE SECOND ORDER NONLINEARITY OF BOOLEAN FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1331-1349 ◽  
Author(s):  
XUELIAN LI ◽  
YUPU HU ◽  
JUNTAO GAO
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order ◽  
Algebraic Degree ◽  
Positive Integers ◽  
Second Order Nonlinearity ◽  
Better Than ◽  
Monomial Functions

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

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.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

scholarly journals Partitioning the positive integers with higher order recurrences

1991 ◽  
Vol 14 (3) ◽  
pp. 457-462 ◽  
Author(s):  
Clark Kimberling
Keyword(s):  
Positive Integer ◽  
Positive Root ◽  
Irrational Number ◽  
Higher Order ◽  
Typical Result ◽  
Algebraic Integers ◽  
Positive Integers

Associated with any irrational numberα>1and the functiong(n)=[αn+12]is an array{s(i,j)}of positive integers defined inductively as follows:s(1,1)=1,s(1,j)=g(s(1,j−1))for allj≥2,s(i,1)=the least positive integer not amongs(h,j)forh≤i−1fori≥2, ands(i,j)=g(s(i,j−1))forj≥2. This work considers algebraic integersαof degree≥3for which the rows of the arrays(i,j)partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): ifαis the positive root ofxk−xk−1−…−x−1fork≥3, thens(i,j)is a Stolarsky array.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

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