Table of Content

Abstract:

The summation of fractional parts is an old topic in number theory since the time of G.H.Hardy and J.E.Littlewood (see [3]). Throughout the years, many mathematicians have contributed to the estimation of the sum \(\sum_{n \leq N} \left\{\alpha n\right\}\) , where α is an irrational number. In Section 2, we estimate the fractional part sum of certain non-linear functions, which can be applied to refine an existing bound of the discrepancy. In Section 3, we continue to make use of the sum in order to study the distribution of quadratic residues and ‘relatively prime numbers’ modulo integers.

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We study the Iterated Circumcentres Conjecture proposed by Goddyn in 2007: Let \(P_1,P_2,P_3,\dotsc\) be a sequence of points in \(R^d\) such that for every \(i \geq d + 2\) the points \(P_i-1,P_i-2,P_i-3,\dotsc,P_i-d-1\) are distinct, lie on a unique sphere, and further, Pi is the center of this sphere. If this sequence is periodic, then its period must be \(2d + 4\). We focus on cases of \(d = 2\) and \(d = 3\) and obtain partial results on the conjecture. We also study the sequence and prove its geometrical properties. Furthermore, we propose and look into several variants of the conjecture, namely the Skipped Iterated Circumcentres Conjecture and the Spherical Iterated Circumcentres conjecture.

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The Erdős-Szekeres conjecture, developed from the famous Happy-Ending Problem, hypothesizes on the number of points in general position needed on a plane to guarantee the existence of a convex n-gon. The research conducted aims to examine geometric characteristics of different constructions of points in general position, organized by number of points forming the convex hull of the set. This paper has explored the case of pentagons, reestablishing the previously proven result of the case using a geometrical approach in contrast to the combinatorial approaches generally adopted when exploring this problem. This paper also proves that the lower bound to the conjecture is not sharp under certain circumstances, an aspect never explored in the past. [See reviewer’s comment (2)]

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In this research, we are interested in how the solutions of the famous Pell’s equation look like. It is well known that the solutions of the Pell’s equation are generated by the fundamental solution of the equation, which could be represented by a set of recursive equations. Therefore, we would like to explore the characteristics of such recurrence sequences and tell the relationship between the cycle length of the congruence modulo a number and divisibility of the terms.

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In this study, we give a synthetic approach to the quadrilateral “Kite”, or more specific to say, the “Right Kite”. It is mainly based on the definitions, postulates (axioms), propositions (theorem and constructions) from the Euclid’s Elements, which is known as one of the most successful and influential mathematical textbook attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c, 300 BC.

Linked up with the definitions of a “Right Kite” and the lines which meet the boundary of a said circle orthogonally described in the Pointcar´e Disk Model, we attempt to combine it in a mathematical task namely “Cryptography”. The application of Poincar´e Disk Model will be acted as a bridge to form a single key for encryption and decryption. Despite the single common trick we use, it leads to infinite possibilities by experiencing various and distinct mathematical skills in cryptography.

Last but not least, we would like to dedicate to the publication of Euclid’s Elements and the discovery of Euclidean Geometry so that we can admire the beauty of Mathematics. Our ultimate goal is to lay the new insight into some of the most enjoyable and fascinating aspects of geometry regarding to the most unaware quadrilateral, Kite

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In this paper, we look into a generalized version of the well-known Tower of Hanoi problem. We will investigate the shortest methods of traversing between any two valid configurations of discs in the standard problem, as well as in some variants.

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In this project, we give an explicit construction of positive defi- nite quadratic forms of arbitrary dimension by using a family of real analytic functions whose coefficients in their Taylor expansions are strictly positive. We also prove a variant result that allows the construction if the number of positive coefficients has a positive upper density.

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In 1903, an anonymous reader submitted a question to Mathematical Questions in The Educational Times: Find all consecutive triples of sums of two squares. J.E. Littlewood later posed a question on whether in general there exist infinitely many triples \(n, n + h, n + k\) that are simultaneously sums of two squares? By solving the equation a \(a^2 + 2 = (a – l)^2 + b^2\), we give all consecutive triples of sums of two squares such that the first number is a perfect square. This method is generalised to solve Littlewoods problem for the case when \(h\) is a perfect square.

We also prove that there are infinitely many pairs of consecutive triples of sums of two squares such that the first numbers of the two triples differ by 8.

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