If p and q are two numbers, each of which is the sum of two squares, then their product pq will also be the sum of two squares. An odd prime pis a sum of two squares if and only if p 1 mod 4. If n is odd, then n is a sum of three squares unless n 7 mod 8. Arithmetic geometry is the study of number theoretic problems informed by the insights of geometryamong them algebraic geometry, topology, differential geometry, and. Since n is odd, an odd number of the three squares must be odd, and so an odd number of these three numbers must be 1. Sums of squares and sums of triangular numbers induced by partitions of 8 article pdf available in international journal of number theory 0404 august 2008 with 1 reads how we measure reads. Number theory, which i taught at queen mary, university of london, in the spring semester of 2009. The sum of two squares problem is an old but nonetheless interesting problem. Despite their ubiquity and apparent simplicity, the natural integers are chockfull of beautiful ideas and open problems. This result is remarkable in that it relates primes. Fermat numbers, mersene like primes are covered in this video. Gazette 94 2010, 119123, with minor variations introduction two of the classic theorems of number theory are. Then x2 1 mod p has a solution if and only if p 2 or p 1 mod 4. Pdf proofs of fermats theorem on sums of two squares.
Therefore, the theorem states that it is expressible as the sum of two squares. A guide to elementary number theory is a 140page exposition of the topics considered in a first course in number theory. It is known that all primes congruent to 1 modulo 4 can be written as a sum of two squares. However, brute force will reveal that 23 cannot be expressed as a sum of three squares. Number theory is a beautiful branch of mathematics.
The journal of number theory features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. In additive number theory we make reference to facts about addition in 1. A primary focus of number theory is the study of prime numbers, which can be. Sums of two squares and lattices keith conrad one of the basic results of elementary number theory is fermats two square theorem. It thus suffices to show that if is a prime of the form, then is a sum of two squares. I rst learned about it in an algebraic number theory course. Fermat numbers, two square theorem final touch for rmo.
A positive integer can be represented as a sum of two perfect squares if and only if in its prime factorization, any prime congruent to. On numbers which are the sum of two squares the euler archive. A combinatorial approach to sums of two squares and. On the other hand, on the other hand, trivially any square n 2 is a sum of two squares we allow for now. Contents preface vii section1integers 1 2uniquefactorization 10 3lineardiophantineequations 20 4congruences 27 5linearcongruences 34 6fermatsandwilsonstheorems 42 7thedivisorsofaninteger 49 8perfectnumbers 56 9eulerstheoremandfunction 63 10primitiverootsandindices 72 11quadraticcongruences 82 12quadraticreciprocity 92 numbersinotherbases 101 14duodecimals 109 15decimals 115. Contents preface vii section1integers 1 2uniquefactorization 10 3lineardiophantineequations 20 4congruences 27 5linearcongruences 34 6fermatsandwilsonstheorems 42 7thedivisorsofaninteger 49 8perfectnumbers 56 9eulerstheoremandfunction 63 10primitiverootsandindices 72 11quadraticcongruences 82 12quadraticreciprocity 92 numbersinotherbases 101 14duodecimals. I discuss here offthebeatenpath beautiful, even spectacular results from number theory.
Fermats two squares theorem states that that a prime number can be represented as a sum of two nonzero squares if and only if or. Results regarding the sum of four squares problem and warings problem are cited with references for further reading. The connection between these numbers and prime numbers will appear later in this article. The well known sum of squares function tells you the number of ways you can represent an integer as the sum of two squares.
The research of the number theory group encompasses classical and algebraic number theory, computational number theory, and especially the modern subject of arithmetic geometry. Representing numbers as the sum of squares and powers in. To see that if n is a sum of two squares then every prime which is 3 mod 4 divides n to an even power we argue by contradiction. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so. Fermats prime theorem on sum of two square number theory. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
In trying to find three squares that sum to 33, we cant use any number whose square is bigger than 33, because once our sum of squares exceeds 33, theres no way to decrease it. Equation 2 follows from the standard set theory formula for the size of the union of two sets. Prime numbers number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. See the link for details, but it is based on counting the factors of the number n into powers of 2, powers of primes 1 mod 4 and powers of primes 3 mod 4. Lagranges foursquare theorem, also known as bachets conjecture, states that every natural number can be represented as the sum of four integer squares.
The prime numbers for which this is true are called pythagorean primes. It turns out to be best to consider first the case of primes. Theres also the sum of two squares theorem for the prime factorization of integers and the pythagorean hypotenuse proposition which says that a number c is a hypotenuse if. Theorem of the day fermatstwosquarestheoreman odd prime number p may be expressed as a sum of two squares if and only if p. More generally, this kind of argument shows that if pis not a sum of two squares, then n pemis not a sum of two squares if eis odd and gcdp. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. The squares are more restrictive, but something good comes from those restrictions. Although he published little, fermat posed the questions and identified the issues that have shaped number theory ever since.
These are questions typical of those which will arise in this course. In additive number theory, fermats theorem on sums of two squares states that an odd prime p can be expressed as. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows. One can replace m and n with two positive natural numbers and obtain a pythagorean triple. Number theory annie xu and emily zhu1 october 5, 2016 1 introduction 1. Theres a theorem that says that p 1 mod 4 if and only if p is a sum of two squares so this number must be at least 1. Find the number of ways sn in which a positive integer n 1 can be represented in the.
Number theory sum of squares and vieta jumping 1 primes 1 and 3 mod 4 the following is an important fact distinguishing primes that are 1 mod 4 and those that are 3 mod 4. Sums of squares, sums of cubes, and modern number theory. We need a lemma which belongs to the theory of quadratic residues see chapter 5. The problem asks which positive integers can be written as the sum of two squares. Sum of squares theorems are theorems in additive number theory concerning the expression of integers as sums of squares of other integers. Warings problem, taxicab numbers, and other sums of powers. Question consider the in nite arithmetic progression. Pdf sums of squares and sums of triangular numbers. In our first lecture we have found the description of all. If one of them is 1, then the other two must be even and add to 6, which is 2 mod 4, while the numbers are 0 mod 4. That is, the squares form an additive basis of order four. The first are those who once knew but have forgotten, for instance, which integers are the sum of two squares and why they have that property, and who want. Every positive integer is the sum of at most four squares. Lectures on analytic number theory tata institute of.