dc.date.accessioned 
20160915T09:45:44Z 
und 
dc.date.accessioned 
20171024T12:22:06Z 

dc.date.available 
20160915T09:45:44Z 
und 
dc.date.available 
20171024T12:22:06Z 

dc.date.issued 
20160915T09:45:44Z 

dc.identifier.uri 
http://radr.hulib.helsinki.fi/handle/10138.1/5744 
und 
dc.identifier.uri 
http://hdl.handle.net/10138.1/5744 

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.discipline.URI 
http://data.hulib.helsinki.fi/id/C3b2c51e946b441e829f14e18bcff245 

ethesis.department.URI 
http://data.hulib.helsinki.fi/id/61364eb4647a40e2853911c5c0af8dc2 

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 
Matematisknaturvetenskapliga fakulteten 
sv 
ethesis.faculty 
Matemaattisluonnontieteellinen tiedekunta 
fi 
ethesis.faculty 
Faculty of Science 
en 
ethesis.faculty.URI 
http://data.hulib.helsinki.fi/id/8d59209f66144edd97441ebdaf1d13ca 

ethesis.university.URI 
http://data.hulib.helsinki.fi/id/50ae46d87ba94821877cc994c78b0d97 

ethesis.university 
Helsingfors universitet 
sv 
ethesis.university 
University of Helsinki 
en 
ethesis.university 
Helsingin yliopisto 
fi 
dct.creator 
Suomalainen, Janne 

dct.issued 
2016 

dct.language.ISO6392 
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 foursquare 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 nonnegative 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 presentday 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 integervalued 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.URI 
http://data.hulib.helsinki.fi/id/languages/eng 

ethesis.language 
English 
en 
ethesis.language 
englanti 
fi 
ethesis.language 
engelska 
sv 
ethesis.thesistype 
pro graduavhandlingar 
sv 
ethesis.thesistype 
pro gradu tutkielmat 
fi 
ethesis.thesistype 
master's thesis 
en 
ethesis.thesistype.URI 
http://data.hulib.helsinki.fi/id/thesistypes/mastersthesis 

dct.identifier.urn 
URN:NBN:fife2017112252385 

dc.type.dcmitype 
Text 
