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Waring's Problem

Show simple item record 2016-09-15T09:45:44Z und 2017-10-24T12:22:06Z 2016-09-15T09:45:44Z und 2017-10-24T12:22:06Z 2016-09-15T09:45:44Z
dc.identifier.uri und
dc.title Waring's Problem en
ethesis.discipline Teaching of Mathematics en
ethesis.discipline Matematiikan opettajan koulutus fi
ethesis.discipline Utbildning av matematiklärare sv
ethesis.department Institutionen för matematik och statistik sv
ethesis.department Department of Mathematics and Statistics en
ethesis.department Matematiikan ja tilastotieteen laitos fi
ethesis.faculty Matematisk-naturvetenskapliga fakulteten sv
ethesis.faculty Matemaattis-luonnontieteellinen tiedekunta fi
ethesis.faculty Faculty of Science en
ethesis.faculty.URI Helsingfors universitet sv University of Helsinki en Helsingin yliopisto fi
dct.creator Suomalainen, Janne
dct.issued 2016
dct.language.ISO639-2 eng
dct.abstract Waring's problem is one of the two classical problems in additive number theory, the other being Goldbach's conjecture. The aims of this thesis are to provide an elementary, purely arithmetic solution of the Waring problem, to survey its vast history and to outline a few variations to it. Additive number theory studies the patterns and properties, which arise when integers or sets of integers are added. The theory saw a new surge after 1770, just before Lagrange's celebrated proof of the four-square theorem, when a British mathematician, Lucasian professor Edward Waring made the profound statement nowadays dubbed as Waring's problem: for all integers n greater than one, there exists a finite integer s such that every positive integer is the sum of s nth powers of non-negative integers. Ever since, the problem has been taken up by many mathematicians and state of the art techniques have been developed - to the point that Waring's problem, in a general sense, can be considered almost completely solved. The first section of the thesis works as an introduction to the problem. We give a profile of Edward Waring, state the problem both in its original form and using present-day language, and take a broad look over the history of the problem. The main emphasis is on the classical version of the problem, whereas the modern version is described in Section 5 with numerous other variations. In addition, generalizations to integer-valued polynomials and to general algebraic fields are described. Goldbach's conjecture is also briefly illustrated. The elementary solution of Waring's problem is presented in Sections 2 to 4. Historical perspective is carried through the thesis with the profiles of the key mathematicians to the solution. The proof presented is an improved and simplified version of Yuri Linnik's solution of Waring's problem. The second section provides the groundwork, an ingenious density argument by Lev Shnirelman, which is applied to the problem in the so called Fundamental lemma presented in Section 3. The proofs of the intermediate results needed to prove the lemma are presented in the following sections. The third section reduces the proof to an estimation of the number of solutions of a certain system of Diophantine equations. The final argument, longish induction is given at the end of the fourth section. Even though Waring's problem is solved, the progress made in the field is far from being idle. The plethora of variations and generalizations, especially Ideal Waring's problem, Modern Waring's problem and Waring–Goldbach problem are actively studied today. It is surprising how deep a problem with such a seemingly simple assertion can be. Conclusively, the challenge in this branch of mathematics is to develop new mathematical methods to prove and explain what seems so obvious. en
dct.language en
ethesis.language English en
ethesis.language englanti fi
ethesis.language engelska sv
ethesis.thesistype pro gradu-avhandlingar sv
ethesis.thesistype pro gradu -tutkielmat fi
ethesis.thesistype master's thesis en
dct.identifier.urn URN:NBN:fi-fe2017112252385
dc.type.dcmitype Text

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