Search

The labeling is said to be lucky labeling of the graph if the vertices of the graph are labeled by natural number with satisfying the condition that sum of labels over the adjacent of the vertices in the graph are not equal, and if vertices are isolated vertex, then the sum of the vertex is zero. The least natural number which labeled the graph is the lucky number. The lucky number of graph \(G\) is denoted by \(\eta \left( G \right)\). The labeling is defined as proper labeling if the vertices of the graph are labeled by natural number with fulfilling the condition that label of adjacent vertices is not the same. The labeling is defined as proper lucky labeling if labeling is proper and also lucky. The proper lucky number of graph \(G\) is denoted by \(\eta_{p} \left( G \right)\). Here, we obtain a proper lucky number for complete bipartite graph \(K_{m,n}\), friendship graph \(F_{n}\) and certain book graph such as triangular book graph \(B_{3}^{t}\) and rectangular book graph \(B_{4}^{t}\).

This game-based activity prompts students to explore the structure of multiplication, experiment with the distributive property, and begin investigating prime numbers.

Estimating the value of numerical superstitions using the hedonic regression method faces an empirical challenge to separate the superstition and non-superstition components. In Malaysia, the number of eight is a lucky number by the Chinese as its pronunciation carries the meaning of “prosperity”; while for the non-Chinese (Malay), it is a neutral number (non-superstitious). This provides a natural setting to use the nonlinear decomposition analysis to remove the non-superstition component from the superstition where the hedonic regression is not able to handle. Using a sample of car plate auction data in Malaysia, this paper estimates the value of superstition in number of eight. The value of superstition is estimated to be RM965 by the hedonic regression. The nonlinear decomposition shows that it is only around 37% to 45% of the estimated superstition value (RM965) can be attributed to superstition. This indicates that the estimation of previous studies on superstitious value of number of eight using the hedonic regression model might be subject to a substantial upward bias.

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.

The labeling is said to be lucky labeling of the graph if the vertices of the graph are labeled by natural number with satisfying the condition that sum of labels over the adjacent of the vertices in the graph are not equal and if vertices are isolated vertex then the sum of the vertex is zero. The least natural number which labelled the graph is the lucky number. The Lucky Number of graph G is denoted by η(G). The labeling defined as proper labeling if the vertices of the graph are labeled by natural number with fulfilling the condition that label of adjacent vertices is not the same. The labeling is defined as proper lucky labeling if labeling is proper and also lucky. The proper lucky number of graph G is denoted by η p (G). Here we obtain a lucky number and proper lucky number for family of quadrilateral snake graph such as quadrilateral snake graph, double quadrilateral snake graph, alternate quadrilateral snake graph and double alternate quadrilateral snake graph.

If l : V ( G ) → N is a vertex labeling of a graph G = ( V ( G ) , E ( G ) ) , then the d-lucky sum of a vertex u ∈ V(G) is d l ( u ) = d G ( u ) + ∑ v ∈ N ( u ) l ( v ) . The labeling l is a d-lucky labeling if dl(u) ≠ dl(v) for every uv ∈ E(G). The d-lucky number ηdl(G) of G is the least positive integer k such that G has a d-lucky labeling V(G) → [k]. A general lower bound on the d-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinite family of corona graphs. The d-lucky number is also determined for the so-called Gm,n-web graphs and graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ is joined to $y$). The {\it additive number} of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a additive labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $\eta_{\ell}(G) $, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $ \eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $. A graph may lack any $(0,1)$-additive labeling. We show that it is $ \mathbf{NP} $-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $ \sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v) $ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $ \varepsilon >0 $, approximating the $(0,1)$-additive number within $ n^{1-\varepsilon} $ is $ \mathbf{NP} $-hard., Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Science

A lucky labeling of a graph G is a function @?:V(G)->N, such that for every two adjacent vertices v and u of G, @?"w"~"v@?(w) @?"w"~"u@?(w) (x~y means that x is joined to y). A lucky number of G, denoted by @h(G), is the minimum number k such that G has a lucky labeling @?:V(G)->{1,...,k}. We prove that for a given planar 3-colorable graph G determining whether @h(G)=2 is NP-complete. Also for every k>=2, it is NP-complete to decide whether @h(G)=k for a given graph G.

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u) S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,...,k} is the lucky number of G, denoted by @h(G). Using algebraic methods we prove that @h(G)=

In this paper, we will consider an arithmetical approach to the Eratosthenes sieve and to the problem of the twin primes.

In the present work we prove a number of results about gaps between consecutive primes. The proofs need the method of Y. Zhang which led to the proof of infinitely many bounded gaps between primes. Several of the results refer to the so-called Polignac numbers which we define as those even integers which can be written in infinitely many ways as the difference of two consecutive primes. Others refer to several 60–70 years old conjecture of Paul Erdős about the distribution of the normalized gaps between consecutive primes and about the distribution of the ratio of consecutive primegaps. The methods involve an extended version of Zhangs method, a property of the GPY weights proved by the author a few years ago and other ideas as well.

While prime numbers are the fundamental building blocks of the integers, understanding how they are spread within the integers has turned out to be hard work. For example, the Prime Number Theorem stood as a conjecture for nearly a hundred years, and anyone who bags the Riemann Hypothesis first will be a million dollars richer. Modern cryptography assumes that the primes will retain their secrets for some time to come. In the presence of hard problems, it is tempting to employ models. A good model should provide an approximation of reality which is simple enough to understand, yet accurate enough to be useful. While there are several models for the primes, in this paper we tell the story of a beautiful and compelling probabilistic model known as the Hawkins primes. First introduced by David Hawkins in this MAGAZINE [11], the model is based on a simple variation of the sieve of Eratosthenes. Over the past fifty years, the Hawkins model has been used to predict the truth, in the strongest probabilistic sense, of results (both established and conjectured) concerning the distribution of the prime numbers, including the Twin Primes Conjecture and the Riemann Hypothesis. Also, the model (or generalizations thereof) has potential to shed light on interesting integer sequences other than the primes.

We study some of the extremely strong statements that can be made about the distribution of primes assuming the (unlikely) existence of exceptional Dirichlet characters. We treat this in general and then apply the results to the particular cases of primes of the form a2 + b6 and of elliptic curves having prime discriminant.

To investigate color and number preferences in Japan, 586 university undergraduates (239 men and 347 women; M age = 20.9 yr.) were asked to name a color (Question 1), to name their preferred color (Question 2), and to name their preferred number between zero and nine (Question 3). The results showed that Japanese students chose blue (33.5%) or red (26.0%) when asked to name a color but that red was not chosen as frequently as blue as a preferred color (red: 11.1%, blue:37.1%). Sex differences were found on both Questions 1 and 2 by chi-squared test. Black was chosen more by men, while pink was selected more by women. 22.5% subjects also selected the number seven, supporting Simon's observation of the “blue-seven phenomenon.” The reasons given for the choice showed that seven was “a lucky number” and “represented happiness” among Japanese students. Four colors (blue, red, white, and black) accounted for 76.8% and 65.1% of responses to Questions 1 and 2, respectively, and odd numbers 68.4% for Question 3.

In this article I present the of demonstration Goldbach conjecture: every pair number is the sum of two primes. As only reference: the logic mathematical and also, the enunciate of Euclides in his proof for the existence of infinitely many primes.

This article brings together three seemingly disparate fields of mathematics: 1) prime numbers; 2) base 6 numbers;3) helix.From a simple sieve of Eratosthenes, based on the base 6 numbers, you go to a three-dimensional structure of the helix, by rolling the plane.

A prime number is a natural number that is not divisible without remainder by any other natural number but 1 and itself. Primes are scattered irregularly among the set of natural numbers, and this fact has fascinated mathematicians throughout the centuries. Today, primes are not only a challenge for mathematicians, but are of very practical value – for instance, they play a central role in cryptography. In this chapter the authors explain a very old, but still practical algorithm for computing a table with all prime numbers up to a specified number.

We establish an asymptotic formula for the number of k -difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59–81]. We here extend this to k ⩾ 2 and generalize it to all l -tuples of Hawkins primes.

However, I have seen the light! Teaching math through problem solving has become my mantra this year and has had a powerful impact on my two fifth-grade math classes. Each class consists of about twenty students ranging in ability from gifted (about 40 percent) to those who require documented-progress monitoring programs because of their low scores on last year's state standardized test (about 25 percent). Approximately 35 percent are typical fifth-grade learners. Students come from traditional, ability-grouped fourth-grade classes and have had little problem-solving experience. Although my approach was met with some resistance from my students ("There are too many words!" or "I just don't get it!"), they have not only come around; they are now enthusiastic problem solvers. Teaching based on problem solving brings challenges for the teacher, primarily that of finding problems with multiple access points that accommodate all students. I found the Four fours problem on a Web site from Bogazici University in Istanbul, Turkey (McQuade). This is a perfect fifth-grade problem. My lucky number has always been four, and the day I presented the Four fours problem to my classes was absolutely my lucky day. Enthusiasm and intrigue filled the air after I introduced the problem. Every student from the most capable to the struggling quickly reached for pencil and paper with a furrowed brow. The problem follows

The Sieve of Eratosthenes has been recently extended by excluding the multiples of 2, 3, and 5 from the initial set, and finding the additive rules that give the positions of the multiples of the remaining primes. We generalize these results. For a given k we let the initial set S k consists of natural numbers relatively prime to the first k primes, and find the rules governing the positions of the multiples of the remaining elements.

For a long time it was believed that sieve methods might be incapable of achieving the goal for which they had been created, the detection of prime numbers. Indeed, it was shown [S], [B] to be inevitable that, in the sieve’s original framework, no such result was possible although one could come tantalizingly close. This general limitation is recognized as the \parity problem" of sieve theory. More recently, beginning with the work [IJ], this goal has in certain cases become possible by adapting the sieve machinery to enable it to take advantage of the input of additional analytic data. There have been a number of recent developments in this regard, for example [H], [DFI], [FI], and several recent works of R.C. Baker and G. Harman. The story however is far from flnished. In this paper we consider a sequence of real nonnegative numbers

We examined the amount of money bet during a week of Pennsylvania’s Daily Number game. In this game, players receive a predetermined payoff for picking the 3-digit number (000 to 999) drawn on that day. The betting distribution was distinctly nonuniform, and we identified several betting patterns, such as picking triples and avoiding double 9s. In addition, we asked separate groups of subjects to rate selected numbers for randomness, luckiness, and perceived history of winning; to categorize numbers; and to free associate to numbers. We propose that people seem to choose highly patterned, available, and/or “lucky” numbers. People apparently do not bet numbers that reflect the random process of the game (they do not utilize a representativeness heuristic).

The results in this paper suggest that Goldbach's conjecture that every odd number is the sum of three primes is true even under the requirement that two of the primes be the same and the third be “arbitrarily” small.

This chapter presents the history of the sieve process. In a sieve problem, two numbers are given A and B, k integers or moduli, k other integers, and a matrix, and the problem is to find an unknown integer x that satisfies an equation. The sieve problem is to find all those integers satisfying the two kinds of conditions, congruential conditions, and inequalities. The chapter discusses that there is another type of sieve problem called quadratic in which n i is approximately equal to ½ m i . This means that a random X has about a 50% chance of satisfying any one of these congruences and if one has k of these the number X has about one chance in 2 k . The sieve method had a relay, some contacts, and circuitry that was very primitive but quite effective. The sieve has the advantage over the stencil method in that one do not have to plan in advance how one wish to go and apply carefully one stencil after another. It was a genuine parallel machine.

The sequence of primes can be defined by the sieve of Eratosthenes. One can think of variations in the definition of a sieve. The following problem was studied by means of actual calculation on an electronic computing machine (partly with the aim of trying to develop "coding procedures" which would obviate the need for large memories which at first appear necessary for problems of this sort). Consider the sequence of all positive integers, 1, 2, 3, e,. . We shall now strike out from this sequence every second term by counting from 1. The odd integers will be left. We shall now strike out every third integer in the remaining sequence, again starting to count from 1, but considering only the rema-ini.ng integers. We shall obtain a second sequence of integers. The next step is to strike out every fourth integer counting only the remaining ones and we obtain another subsequence. We can continue this process Indefinitely. It is obvious that infinitely many integers will remain after we have completed the process. This sieve is different from that of Eratosthenes since in striking out all multiples of successive integers we count off only among the remaining ones. It could perhaps be called a sieve of Josephus Flavius.The result of this sieve is a sequence of integers of a density much smaller than that of the primes. Another sieve could be the following: consider again the sequence of integers starting with 1. We shall strike out from it every second term.Apart from 1, the first integer which remains is 3; now in the remaining sequence we shall strike off every term whose index is a multiple of 3. In the sequence which remains now and which consists of 1, 3, 7, 9, 13, 15, 19, ..., the first term which has not been used already is 7. We shall,therefore, strike off every term in this sequence whose index is a multiple of 7; that is to say, the 7th (which is 19),, 14th, 21st, etc., term in this sequence. In the remaining sequence we shall look up the first term which has not been used (it is 9) and again strike off terms whose indices are multiples of it. We continue this process indefinitely; it is obvious again that infinitely many terms remain. They may be called the result of our sieve. The aim of our exercise was to consider certain asymptotic properties of this latter sequence of numbers, let us call them for brevity "lucky numbers." All lucky numbers up to 48,000 were quickly computed on the machine and the following data about them were obtained.(For the first few see Table I.)

The lucky numbers of Ulam resemble prime numbers in their apparent distribution among the natural numbers and with respect to the kind of sieve that generates them [11. It is therefore of interest to investigate their properties, bearing in mind the analogies to prime number theory. The lucky numbers are defmed by the following sieve. If Sn is an infinite sequence of natural numbers tn (m = 1,2,53, ...), one obtains Sn reovin evra for which t divides Sn. for n > 1, from Sn by removing every tn a o hc tn,ndide m. S2 is the sequence 2,3,5,7,9, ... of the number 2 followed by the odd integers in increasing order of magnitude. Si is the seqence of natural numbers. The -sequence of lucky numbers is S =lim Sn , that is