Niven completed the solution of most of warings problem in 1944. By the way, before turning to the modern form of warings problem, it is perhaps. Thus, as a consequence of work of deshouillers, hennecart and. Many of the most perplexing problems in number theory arise from. Some famous problems of the theory of numbers and in particular warings problem. This has its genesis in a celebrated paper of hardy and ramanujan 64 devoted to the partition function. Many of the problems are mathematical competition problems from all over the world like imo, apmo, apmc, putnam and many others. Additive number theory had its origin exactly 300 years ago in fermats assertion that he was the first to discover the gen eral theorem that every integer is a sum. Warings problem, taxicab numbers, and other sums of powers. It covers the basic background material that an imo student should be familiar with. Neale easter term 2012 course description this course is intended as an introduction to one of the central topics of additive number theory, namely warings problem and its solution using the hardylittlewood circle method. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Warings problem, taxicab numbers, and other sums of powers 8 dec 2004 1. In number theory, warings problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k.
Introduction to number theory lecture notes adam boocher 20145, edited by andrew ranicki 20156 december 4, 2015 1 introduction 21. Apostol, nitro pdf free merge introduction to analytic number theory, narosa publishers, 1998. Paul halmos number theory is a beautiful branch of mathematics. The paper deals with two elementary methods for solving warings problem on the representation of numbers as a sum of equal exponents in powers of natural numbers. Recall that the usual waring problem requires all positive signs. Buy some famous problems of the theory of numbers and in particular warings problem on free shipping on qualified orders. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. Some famous problems of the theory of numbers and in. Hua loo keng, the method of trigonometric sums and its applications in number theory. By the way, before turning to the modern form of warings problem, it. The course is tentatively expected to cover the following topics. Warings problem has a very long history with an extensive literature spanning many different areas of mathematics. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics.