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MATH 361: NUMBER THEORY — FIRST LECTURE1. IntroductionAs a provisional definition, view number theory as the study of the properties ofthe positive integers,Z+= {1, 2, 3, · · · }.Of particular interest, consider the prime numbers, the noninvertible positive inte-gers divisible only by 1 and by themselves,P = {2, 3, 5, 7, 11, · · · }.Euclid (c.300 B.C.) and Diophantus (c.250 A.D.) posed and solved number theoryproblems, as did Archimedes. In the middle ages the Indians and perhaps theChinese knew further results. Fermat (early 1600’s) got a copy of Diophantus andrevived the subject. Euler (mid-1700’s), Lagrange, Legendre, and others took itseriously. Gauss wrote Disquisitiones Arithmeticae in 1799.Example questions:• (The perfect number problem.) The numbers 6 = 1 + 2 + 3 and 28 =1 +2 + 4 + 7 + 14 are perfect numbers, the sum of their proper divisors. Arethere others? Which numbers are perfect? This problem is not completelysolved.• (The congruent number problem.) Consider a positive integer n. Is there arational right triangle of area n? That is, is there a right triangle all three ofwhose sides are rational, that has area n? This problem is not completelysolved, but very technical 20th century mathematics has reduced it to aproblem called the Birch and Swinnerton-Dyer Conjecture, one of the so-called Clay Institute Millennium Problems, each of which carries a milliondollar prize. The book Introduction to Elliptic Curves and Modularforms by Neal Koblitz uses the congruent number problem to introducethe relevant 20th century mathematics.• (A representation problem.) Let n be a positive integer. Which primes ptake the formp = x2+ ny2for some x, y ∈ Z? This problem is solved by modern mathematical ideas,specifically complex multiplication and class field theory. The book Primesof the Form x2+ ny2by David Cox uses the representation problem inits title to introduce these subjects.• (Questions about the distribution of primes.) Are there infinitely manyprimes? Infinitely many 4k + 1 primes? Infinitely many 28k + 9 primes?If so, can we say more than infinitely many? If π(x) denotes the numberof primes p ≤ x then how does π(x) grow as x grows? Questions like thislead quickly into analytic number theory, where, for example, ideas from12 MATH 361: NUMBER THEORY — FIRST LECTUREcomplex analysis or Fourier analysis are brought to bear on number theory.It is known that asymptoticallyπ(x) ∼ x/ ln(x)but the rate of asymptotic convergence depends on the famous Riemannhypothesis, another unsolved problem.• (The Goldbach Conjecture.) Is every even integer n ≥ 4 the sum of twoprimes? This problem is unsolved.For all of these problems, the answer is either unknown or requires mathematicalstructures larger than Z+and its arithmetic. On the other hand, plenty can bedone working entirely inside Z+as well.So, loosely speaking, there are two options for a first course in number theory,elementary or nonelementary. (Here elementary doesn’t mean easy, but ratherrefers to a course set entirely in Z.) And again speaking loosely, a nonelementarycourse can be algebraic or analytic. This course will be nonelementary, with moreemphasis on algebra than on analysis, although I hope to introduce some analyticalideas near the end of the semester to demonstrate their interaction with the algebra.Despite the emphasis on algebra in this class, the abstract algebra course is notprerequisite. This course is equally a good venue for practicing with algebra or forbeginning to learn it.Our text, by Ireland and Rosen, is well-suited to the emphasis of the course. Wewill cover its first nine chapters and a selection of its later material.Many books are on reserve for this course as well, e.g., Cox, Hardy and Wright,Koblitz, Marcus, Silverman and Tate, Niven and Zuckerman and Montgomery, andso on. Many of these books are mostly about subjects that we will only touch on.Feel welcome to come see me for guidance about reading beyond the course.2. The Gaussian IntegersAs an example of an algebraic structure larger than the integers, the ring ofGaussian integers isZ[i] = {a + ib : a, b ∈ Z},with its rules of addition and multiplication inherited from the field of complexnumbers. The Gaussian integers form a ring rather than a field, meaning thataddition, subtraction, and multiplication are well-behaved, but inversion is not:the reciprocal of a Gaussian integer in general need not exist within the Gaussianintegers.As a ring, the Gaussian integers behave similarly to the rational integers Z. Theunits (multiplicatively invertible elements) of the Gaussian integers areZ[i]×= {±1, ±i},a multiplicative group. Every nonzero Gaussian integer factors uniquely (up tounits) into prime Gaussian integers. And so on.Prime numbers in Z are called rational primes to distinguish them from primenumbers in the Gaussian integers. The somewhat awkward phrase odd prime meansany prime p other than 2.MATH 361: NUMBER THEORY — FIRST LECTURE 33. Prime Sums of Two Squares via the Gaussian IntegersTheorem 3.1 (Prime Sums of Two Squares). An odd rational prime p takes theform p = a2+ b2(where a, b ∈ Z) if and only if p ≡ 1 mod 4.(Note: The notation p ≡ 1 mod 4 means that p = 4k + 1 for some k. In general,the language x is y modulo n means that x and y have the same remainder upondivision by n, or equivalently, that n divides y − x.)Proof. ( =⇒ ) This direction is elementary. An odd rational prime p is 1 or 3modulo 4. If p = a2+ b2then p ≡ 1 mod 4 because each of a2and b2is 0 or 1modulo 4.( ⇐= ) This direction uses the Gaussian integers. For now, take for granted afact about their arithmetic:p ≡ 1 mod 4 =⇒ p factors in Z[i].Granting the fact, we havep ≡ 1 mod 4 =⇒ p factors in Z[i]=⇒ p = (a + ib)(c + id), a + ib, c + id /∈ Z[i]×=⇒ p2= pp = (a2+ b2)(c2+ d2)=⇒ p = a2+ b2= c2+ d2.So the arithmetic of the Gaussian integers has made the problem easy, but nowwe need to establish their arithmetic property that p factors in Z[i] if p ≡ 1 mod 4.To do so structurally, we quote a fact to be shown later in this course,{1, 2, 3, · · · , p − 1} = {1, g, g2, · · · , gp−2} for some g, working modulo p.That is, some element g of the multiplicative group {1, 2, 3, · · · , p − 1} (again,working modulo p) generates the group. Equivalently, the group is cyclic. Thegenerator g need not be unique, but choose some g that generates the group, andleth = g(p−1)/4.The definition of h is sensible


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