Table of Content

Abstract:

In this paper, we are going to investigate the ${\bf{\textit{Erdős-Straus Conjecture }}}$: For any positive $n \geq 2$, there exists positive integers $k,k_1,k_2$ such that
$$ \dfrac{4}{n} = \dfrac{1}{k}+\dfrac{1}{k_1}+\dfrac{1}{k_2}$$

Firstly, we will solve a simpler form $ \dfrac{3}{n} = \dfrac{1}{x} + \dfrac{1}{y}$ as a starting point. Next we will investigate the Erdős-Straus Conjecture in the following dimensions: the related geometric representation of the Erdös-Straus Conjecture, the properties of solutions of the Erdős-Straus Conjecture, further investigation of some paper of the Erdös-Straus Conjecture, existence of special forms of solutions of the Erdős-Straus Conjecture, and the investigation of the Erdős-Straus Conjecture in algebraic dimension. The aim of this report is to find evidence that shows the ${\bf{\textit{Erdős-Straus Conjecture}}}$ is true. If evidence is not strong enough, we still hope that this report can make an improvement to the researched result at present.

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Abstract:

The problem of finding all integral side length of a right-angled triangle is famous and the solution set is called the Pythagorean triple.

Now, instead of sides of a triangle, we concern the orthogonality of lines from vertices to their opposite sides. We want to generalize the problem to arbitrary rational ratio on the sides.

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Abstract:

In this project, we have achieved various results using Probabilistic methods. By exploiting the concept of probability and expected value, we manage to achieve three results: on distribution of entries on a cube, on colouring of vertices of a hypergraph and a lower bound of a maximal independent set on a hypergraph.

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Abstract:

The traditional sieve of Eratosthenes gives a simple algorithm for finding all prime numbers. However, prime numbers seem appear unpredictably but with regular population ratio in the ranges of integers, as Gauss found a density function of prime numbers within a range of x. On the other hand, there are a few methods of classification of prime numbers. We developed a new classification of prime numbers by prime number trees. In the prime number trees, the followed number is generated by attaching a digit either 1, 3, 7, or 9 to the right hand side of the preceding prime number. If the number generated remains prime, then the process is continued, otherwise it is stopped. The prime number trees group prime numbers with similar digits together and show the elegancy of a shorthand of prime numbers. This method also shows a regular classification of prime numbers.

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Abstract:

In this paper, we look into the (m, n, k, p, q) game, one of the generalizations of the well-known Tic-Tac-Toe game. The objective of the game is to achieve ‘k-in-a-row’ with one’s pieces before one’s opponents does. We use two methods — exhaustion and pairing strategy — to investigate the results of the (m, n, k, p, q) game for several different values of the five parameters.

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Abstract:

The paper places much emphasis on the method of, without using division, checking the divisibility of an integer by an odd divisor. In part A, it mainly focuses on getting the general way to perform the divisibility test by an algorithm using the unit digit and the rest of truncated digits of the dividend. Parts B and C are extensions of part A. In part B, it attaches the importance on using the last two or more digits of the dividend and so the divisibility test is not just restricted to the ones digit. While parts A and B direct at the method of verifying the divisibility of a number, part C mainly concentrates on finding out the quotient without performing division algorithm. This unique method of division is discovered in the process of investigation in part A.

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Abstract:

This study centres on the Euclidean construction of triangles from several given preconditions, and carries several major objectives surrounding this aim:

1. Devising a scheme to primarily distinguish cases in which Euclidean construction is impossible;
2. Seeking the simplest agenda in the construction of possible cases;
3. Giving a strict definition of Euclidean constructability;
4. Determining methods of rigorously proving inconstructability

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Abstract:

In this project, we establish the Sudoku graph by studying the relationship between Sudoku and graphs with the help of NEPS (Non-complete Extended P-Sum). The approach is to look for the chromatic polynomial of the Sudoku graph, so that we can find out the total number of possible solved Sudoku puzzles. Though the chromatic polynomial of the Sudoku graph is not presented in this research, we have found some properties of the polynomial that may provide inspirations for further researches.

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