Number Theory

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Welcome to the Topic of Number Theory
Part of the School of Olympiads

"Mathematics is the queen of the sciences and number theory is the queen of mathematics." —Gauss

This topic is about Number Theory, or in simple language study of properties of integers. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated.

The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves).

The first chapter Factorisation of Numbers in this Topic basically deals with extensions of Elementary Number Theory or the study on factorizations, and covers topics like Multiplicative Combinatorial Number Theory, Prime Numbers, Perfect Numbers and other Special Numbers, Modular Arithmetic, Arithmetic Functions, Algebraic Integers, and Theory of Reciprocities. Actually for most Number-Theorists, this topic covers most of the subject.

The second chapter Diophantine Analysis is devoted to the study of Diophantine Equations, which are indeterminate equations for which integer solution or class of solutions is asked. Many equations such as Numeral Systems, Pell's Equation, Pythagorean Theorem, Integer Functions and Fixed Points, and Exponential Equations are discussed along with techniques such as Modular Residues, Factoring Expressions, Infinite Descent, Viete Root Flipping and Chakravala are discussed.

The third chapter Playing with Numbers discusses systems that change their state at discrete intervals in strict accordance to predefined rules like Nim Games, Erase & Replace Systems, Solitaires, etc. This topic basically aims to study Arithmetic Mathematical Games through techniques such as Invariants and Infinite Descent. This topic lacks a formal extent in terms of Theorems and is generally considered as part of Recreational Mathematics, although it does carry many good problems.

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Some high-level problems: from respective chapters:

  • Factorisation of Numbers Prove that there are infinitely many positive integers n such that n2+1 has a prime divisor greater than 2n+√2n. [IMO 2008]
  • Diophantine Analysis The positive integers a and b are such that the numbers 15a + 16b and 16a−15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? [IMO1996]
  • Playing with Numbers To each vertex Pi (i = 1, . . . , 5) of a pentagon an integer xi is assigned, the sum s = Σxi being positive. The following operation is allowed, provided at least one of the xi’s is negative: Choose a negative xi, replace it by −xi, and add the former value of xi to the integers assigned to the two neighboring vertices of Pi (the remaining two integers are left unchanged). This operation is to be performed repeatedly until all negative integers disappear. Decide whether this procedure must eventually terminate. [IMO 1986]

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