Pell's Equation and Fundamental Units Pell's equation was first introduced to me in the number theory class at Caltech that I never comleted. It was r
|
|
- Timothy Boone
- 5 years ago
- Views:
Transcription
1 Pell's Equation and Fundamental Units Kaisa Taiale University of Minnesota Summer 000 1
2 Pell's Equation and Fundamental Units Pell's equation was first introduced to me in the number theory class at Caltech that I never comleted. It was resented in the context of solving iohantine equations - finding integral solutions to equations. It looks easy: find the smallest ositive integer solutions (x; y) to the equation x y = 1, with a non-square ositive integer. But althought there are several algorithms for finding solutions, a attern in the solutions has never been detected. Moreover, values of which one might exect to roduce slightly different solutions can at times vary dramatically. A story might illustrate: One day Iwas sitting haily at my comuter, answering , when I got a note that said merely, The smallest non-trivial integer solution of x 94y =1isx = and y = 1064:" This intrigued me, for some reason, and I asked what some other solutions were. Iwas urged to ick anumber for, and I icked 151. Solving these equations using a somewhat-better-than-brute-force method took one minute for = 94 and three hours for = 151, finally resulting in x = and y = Before the solution was found I started to wonder why itwas taking so long, and why I'd want towait so long to hear those numbers, and whether or not there was any way to estimate the size of solutions so that I'd know whether or not I wanted to embark on any such searches in the future. In resonse, I was handed a aer by Yoshihiko Yamamoto, ublished in Titled, Real Quadratic Number Fields with Large Fundamental Units," it seemed to have nothing to do with Pell's equation. Uon the asking of some questions, like, What's a fundamental unit?" and some research, the connection began to get clearer. Although at first glance the fundamental unit has little to do with the equation I saw inmy number theory class it turns out that they are intimately connected. The fundamental unit ffl of the ring of algebraic integers in a real quadratic number field is a generator of the grou of units (mod ±1). For the subring Z[ ] (consisting of elements x + y with x; y Z) of the ring of integers in Q( ), ffl is equal to x + y where (x; y) is the smallest integer solution for Pell's equation. So finding bounds for the size of the fundamental unit is extremely useful, as when one knows ffl and one can find bounds for x and y. Yamamoto's aer does not address the general subject of finding bounds for ffl. In his introduction, he remarks uon the fact that there are many works dealing with the estimation of the fundamental unit, but that they rimarily deal with fundamental units with small orders of absolute value in comarison to their discriminants." His aer deals instead with constructing real quadratic number fields with comaratively large fundamental units. The essence of the work is that one can find a ositive constant c 1 such that log ffl>c 1 (log( ) for sufficiently large. After the introduction, the aer is divided into four arts. In the first, Yamamoto outlines the idea of reduced quadratic irrationals. In the second, he links these to reduced ideals, which he also defines. Only in the third section does he directly aroach the toic of bounds for ffl. The fourth consists of examles. In outlining Yamamoto's research I will roughly follow his format, adding exlanation or other materials wherever necessary. The first thing Yamamoto addresses is reduced quadratic irrationals. Begin with a quadratic equation, ax + bx + c = 0, a > 0 and gcd(a; b; c) = 1. Then let ff be a root of this equation and let be the discriminant, equal to b 4ac. Then ff is a real number, the root of a quadratic equation, and irrational, because the here is the same as the of Pell's equation. Thus the name quadratic irrational. There exists, too, a conjugate to ff, denoted ff 0. We call ff reduced if ff>1 and 0 >ff 0 > 1. Every quadratic irrational is equivalent to a reduced quadratic irrational, where equivalence between two quadratic irrationals ff and fi is defined by the relation ff = afi + b cfi + d : Here a; b; c; d are integers and ad bc = ±1. The equivalence of ff and fi means that they have the same discriminant.
3 In addition to satisfying the conditions rovided above, a quadratic irrational is deemed reduced if its continued fractional exansion is strictly eriodic. The continued fractional exansion is found as follows: ff = ff 1 ff i = a i + 1 ff i+1 Here a i is the largest integer not larger than ff i. If ff N+1 = ff the exansion has eriod N; if this is true for no N then ff is not eriodic. An examle: for = 1, is a reduced quadratic irrational. Not only do the inequalities hold > 0 and 0 > 1 3 > 1 but the eriodic exansion of 1 + 3is remarkably easy to find ff 1 =1+ 3= ff = = ff 3 = ff 1 The quadratic irrational numbers can be organized into sets based on the revious information. The most basic division is to grou them based on whether or not they are reduced. We will use A Λ = A Λ () to mean the grou of all quadratic irrationals and A = A() to mean the grou of reduced ones. Beyond this, one can use the continued fractional exansion of ff to divide A into cosets. The continued fractional exansion of ff results in the sequence of numbers ff i (i=1,,..., N), and each of these sequences makes u a coset of A with resect to the equivalence relation. Let A = A 1 [ A [ ::: [ A h be the equivalence class decomosition of A, then the number h of the cosets is equal to the ideal class number of the field F = Q( )if is the discriminant of F." To me, this is a very interesting result. I had never seen even the hrase class number" before reading this aer, and so I decided I should look it u. It turns out that the result just exlained above is similar to avery classic way of finding the class number - John Stillwell exlained class number in the introduction to irichlet's Lectures on Number Theory" as [t]he number of inequivalent forms with a given discriminant." He was talking about binary quadratic forms (ax +bxy + cy = n, n an integer). Equivalent forms are forms for which a certain change of variables (x 0 = ffx + fiy, y 0 = flx + ffiy, ffffi fifl = ±1, ff; fi; fl; ffi all integers) does not change the values a form takes. An examle of two inequivalent forms are those with = 5, x +5y and x +xy +3y. These two exressions do not take the same values. There are many arallels between this way of finding class number and the secific method used above. Most recisely, the class number is the cardinality oftheideal class grou, which is the quotient grou G=S where G is the grou of all non-zero fractional ideals, and S is the subgrou of all non-zero rincial fractional ideals. (These terms are defined later in this aer.) But another, erhas more intuitive, wayof thinking of class number is that it is very roughly the number of ways an algebraic integer can be factored in the ring of algebraic integers in the field Q( ). This brings us back to our real quadratic number field, and we can sto thinking about x, y, and changes of variable. It also makes more aarent the idea of class number as a measure of failure of unique factorization in a ring. This is intriguing idea, esecially for those who are just venturing into the world of number theory and have never thought of the imortance of unique factorization. Even more interesting is a simle relationshi that exists between class number and fundamental unit: hffl has a constant value. Yamamoto does not use this result. It is startling enough to this author, though, that it may beanavenue of further inquiry. But back towork. From ff we can also find ffl. It turns out that for ff A, ff = aff + b cff + d Then ad bc =( 1) N, where N is the eriod of the continued fractional exansion of ff, and ffl = cff + d. There are several other ways to find ffl from ff, though. Proosition 1. in Yamamoto's aer says, 3
4 Ifisequal to the discriminant of a real quadratic number field F and ffl is the fundamental unit of F, then Y ffffla i ff = ffl for any equivalence class A i (i =1; ; :::; h). Corollary 1.3 says that in addition to that, Q ffffla ff = fflh. The corollary follows fairly easily from the roosition, and both can be roven by inductive reasoning. The next section in Yamamoto's aer addresses the relationshi between reduced quadratic irrationals and what will be called reduced ideals. Although the first section was mathematically accessible using only high-school algebra, the second section required me to introduce myself to abstract algebra, as well as a few more things. The first thing Yamamoto brings u is!. Let! = +, and notice that 1 and! make u a basis of the ring R of all algebraic integers in F, where F = Q( ) is the real quadratic number field F with discriminant. Then every integral ideal I has the form [a; b + c!] with these conditions: a; b; c Z a>0;c>0;ac= N(R) a = b = 0 mod c and N(b + c!) =0modac a <b+ c! 0 < 0 where! 0 is the conjugate of!. Then ff=ff(i) = b+c! a. We call ff the quadratic irrational associated with the ideal I, and I is reduced if c=1 and ff(i) is a reduced quadratic irrational. A brief review of terms: An ideal I of a ring R is a subset of R for which, if r R, ri = fraja Ig and Ir = farja Ig. In addition, I is closed under multilication by elements from R. In other words, if any element in a ring is multilied by an element of one of the ring's ideals, their roduct is an element of the ideal. A quick examle is the ideal generated by in the ring of integers: multily any integer by two and it's obvious that the roduct is a member of the set of even integers. Yamamoto refers to integral ideals when seaking of these subsets; this is to distinguish them from fractional ideals. Although never exlicitly mentioned, fractional ideals are necessary in the definition of ideal class grou, for instance, so here is a definition: Let o be the ring of algebraic integers in Q( ). Then a fractional ideal of o in Q( ) is a non-zero finitely-generated o-module inside Q( ). (What does that last art mean? An o-module M is finitely generated if there is a finite list m 1 ;:::;m n of elements in M so that M = om om n. A module is to a ring as a vector sace is to a field.) An integral ideal, then, is just a fractional ideal contained in o. Yamamoto also writes of rincial ideals rincial simly means that all the elements in the ideal are multiles of the generator. The main roosition of section two of the aer is this (Proosition.1): The ma I! ff(i) gives a bijection of the set of all reduced ideals to the set A = A() of all reduced quadratic irrationals with discriminant. And it induces a bijection of the ideal class grou of F to the set A 1 ;A ; :::; A h of the equivalence classes of A. Of slightly less theoretical imort, but extremely useful in the roof of the theorem the aer is aiming at, is Proosition.: An integral ideal I is reduced if(i)n(i) < and (ii) the conjugate ideal I 0 is relatively rime to I. 4
5 Once again, a definition of terms: If an ideal has the form [a; b + c!], then its conjugate has the form [a; b + c! 0 ]. For two ideals to be relatively rime means that only roducts of their elements are in their intersection. These two roositions are fairly significant. The technique of associating ideals with numbers is fairly common, and a good tool. Since numbers are often easier to deal with than ideals, a bijection of a set of ideals to a set of numbers can be a good way to turn an algebra roblem into a roblem with numbers rather than ideals. However, in this roof, we'll see that Theorem 3.1 deals with ideals, and that since the bijection exists the result holds for actual numbers. Section three of Yamamoto's aer brings all the revious information together into a roof of the theorem given at the beginning. What follows, in my aer, is essentially a commentary on and exlanation of the roof given in Yamamoto's note. Theorem 3.1. Let i (i = 1; ; :::; n) be rational rimes satisfying 1 < < < n. Assume that there exist infinitely many real quadratic number fields F satisfying the following condition (*): (*) Every i is decomosed inf into the roduct of two rincial ideals m i and m 0 i. Then there exists a ositive constant c 0 deending only on n and 1 ; ; ; n such that log ffl>c 0 (log ) n+1 holds for sufficiently large, where and ffl are the discriminant and the fundamental unit for F. Proof. Consider the ideals I of the form I = ny i=1 m ei i m0fi i Then I is a rincial integral ideal and reduced if(a)n(i) = e1+f1 1 en+fn n and (b) e 1 f 1 = = e n f n =0(Proosition.). < This is not hard to see: Proosition. says that an integral ideal I is reduced if N(I) <, and it is easy to see that (a) follows this exactly. Although slightly less obvious, (b) follows from art (ii) of Pro... Condition (b) makes it necessary that either e i or f i is equal to 0, for any i. This ensures that I and I 0 are always relatively rime to each other. Let I 1 ;I ; ;I t be the set of all reduced ideals obtained as above. Then the quadratic irrationals ff 1 ;ff ; ;ff t associated with them build a subset of the equivalence class A 1, say, corresonding to the rincial ideal class. The rincial ideal class is the collection of fractional ideals which are rincial. Most imortant in that definition is the fact that it is a subgrou of the grou of all fractional ideals. This allows us to understand the next sentence: So we get, from Proosition 1., ffl = Y ffffla 1 ff> ty i=1 ff i : Proosition 1. says that ffl is the roduct of all the ff in A 1. This roduct is larger than the roduct of all ff i (i =1; ; ;t) because the ff i make u a subset of A 1. Since all ff are larger than one by defintion, multilying more of them makes a bigger number. 5
6 On the other hand we have ff i = b i +! N(I i ) > ( )(e1+f1 1 en+fn n ); where I i =[N(I i );b i +!] is the canonical basis of I i. This basis can be found from section two by noting that if I has basis [a; b + c!], a condition on a is that N(I) =ac, and I is reduced if c=1. Thus the basis can be written as [N(I);b+!]. Hence we get the following inequality (3:1) ffl> 0Y = e1+f1 i en+fn n = ffl 0 : The roduct in (3.1) is taken over all integers e i and f i satisfying (a 0 ) (e 1 + f 1 ) log 1 + +(e n + f n ) log n < log( (b 0 ) e i 0;f i 0 and e i f i =0(i =1; ; ;n): ); Conditions (a 0 ) and (b 0 ) follow easily from conditions (a) and (b) given earlier in the roof for (a 0 ) use the laws for logarithms, for (b 0 ) notice that if ab =0,a; b both real numbers, a =0orb =0. The condition that e i and f i be ositive is new but not surrising. We have (3:) 0Y =( )t The number t equals to the cardinal of the set of n-tules (e 1 ;f 1 ; ;e n ;f n ) satisfying (a 0 ) and (b 0 ). Remember that t first came u as the number of ideals I obtained by multilying factors of rimes - check the beginning of the roof. Then it holds (3:3) t = n V P + O((log n 1 ); where V is the volume of the n-simlex in the n-dimensional euclidean sace R n ; =f(x 1 ; ;x n ) R n : x 1 0; ;x n 0; x x n» log( and P = (log 1 )(log ) (log n ): ) Notice that (a 0 )isvery similar to the second descritor for the set, and that all the x i must be greater than or equal to 0. What Yamamoto is doing is saying that a arallel can be drawn between e i and f i and the x i, so that standard results for n-simlices can be alied to this roblem. It would robably be a good idea, at this oint, to exlain what an n-simlex is. A 1-simlex is a line. A -simlex is a triangle. A 3-simlex is a tetrahedron. The attern goes on like this, with n indicating the dimension of the shae with triangular 'sides.' The volume of the n-simlex aroximates the number of lattice oints inside - the number of e i and f i that satisfy the conditions given. Then the O(( )n 1 ) term is an error term. For sufficiently large, the n-simlex estimation is fairly accurate. When all the vertices of an n-simlex have coordinates (0; ; 1; ; 0), in R n, then the n-volume of the simlex is 1 n!. However, in this case the coordinates of the vertices x i have a condition imosed on them: 6
7 x x n» (log ). Yamamoto combines these facts next. We have V = Z dx 1 dx n = 1 n! (log )n : Almost there... For the roduct of all denominators in the right side of (3.1), we have (3:4) log Π 0 ( e1+f1 1 en+fn n ) = = n P = Z 0X [(e1 + f 1 ) log 1 + +(e n + f n ) log n ] (x x n )dx 1 dx n + O((log n n (n + 1)!P (log ) n+1 + O((log ) n ): ) ) This is a fairly easy-to-follow set of maniulations, using only logarithm laws, integration, and algebra. From (3.3) and (3.4), we get [3:45] log ffl 0 = log( = )t log Π 0 ( e1+f1 1 en+fn n ) n (n + 1)!P (log ) n+1 + O((log ) n ): Our theorem follows from this and (3.1). Recall that (3.1) says ffl > Q 0 = e 1 +f 1 en+fn i n n (n+1)!p = ffl 0 : Take the log of both sides, then substitute the exression in [3.45] for log ffl 0. is c 0 in the theorem, and for sufficiently large the error term in [3.45] is negligible. Thus Theorem 3.1 is roven. Theorem 3. is the last theorem in Yamamoto's aer, and I will include it for the sake of comleteness. Not much comment is needed; anything difficult to understand I leave, as an exercise, to the reader. Theorem 3. For the case n =, the assumtion of Theorem 3.1 is satisfied by the following F 's: F = Q( m k ) where we set = 1 and q =. m k =( k q + +1) 4 (k =1; ;:::); Proof We easily see that m k =1(mod ), k =( 1) (mod q) and m k =1(mod 4) (m k =1(mod 8) if =. Hence each one of and q is decomosed into the roduct of two distinct rime ideals in F (if m k is not a square). Set = mm 0 and q = nn 0. From the definition of m k, it holds (3:5) ( k q + +1) m k =4 (3:6) ( k q + +1) m k = 4 k q: 7
8 From (3.5), m and m 0 are both rincial (set m =( k q++1+ m k ), for examle.) From (3.6), either m k n or m 0k n is rincial. Since m k and m 0k are rincial, both n and n 0 are also rincial. So the condition (*) in Theorem 3.1 is satisfied. Finally, the infiniteness of the number of F 's given above is as follows. Set k =j (we consider the case where k is even), then m k = m j =( j q + +1) 4 = q 4j + nq( +1) j +( 1) : Since the diohantine equation y = q x 4 +q( +1)x +( 1) has only a finite number of rational integral solutions (x; y) for a fixed integer (Siegel's theorem), Q( m j ) reresents infinitely many real quadratic number fields F for j =1; ;:::. This comletes the roof. So the theorems about bounds for fundamental units in real quadratic number fields have been roven. I can find, if I want, a lower bound for the fundamental unit ffl in a real quadratic number field with a given large discriminant ; from this I can find some estimate for the solution to Pell's equation with the same. Let us return, then, to the original question: How long is it going to take to find that solution to x + 151y =1? How big is it going to be? And the answer is this: although we nowhave a method for finding a lower bound for ffl, the method itself takes imlementation. It is necessary to ascertain how many rimes smaller than 151 decomose in Q( 151) into the roduct of two rincial rime ideals. This is not the easiest of tasks I could write a comuter rogram to do it for me, but if I am willing to do that then I might as well write a comuter rogram to find the solutions to Pell's equation directly, and erhas use a more efficient algorithm so that it doesn't take three hours. So here my exloration stoed. My summer has ended, and so has the theoretical art of the question I investigated. Now it's all alication... 8
Sets of Real Numbers
Chater 4 Sets of Real Numbers 4. The Integers Z and their Proerties In our revious discussions about sets and functions the set of integers Z served as a key examle. Its ubiquitousness comes from the fact
More informationAlmost 4000 years ago, Babylonians had discovered the following approximation to. x 2 dy 2 =1, (5.0.2)
Chater 5 Pell s Equation One of the earliest issues graled with in number theory is the fact that geometric quantities are often not rational. For instance, if we take a right triangle with two side lengths
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationMATH 361: NUMBER THEORY EIGHTH LECTURE
MATH 361: NUMBER THEORY EIGHTH LECTURE 1. Quadratic Recirocity: Introduction Quadratic recirocity is the first result of modern number theory. Lagrange conjectured it in the late 1700 s, but it was first
More information(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury
Is e π 163 odd or even? (Worksho on Harmonic Analysis on symmetric saces I.S.I. Bangalore : 9th July 004) B.Sury e π 163 = 653741640768743.999999999999.... The object of this talk is to exlain this amazing
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More information3 Properties of Dedekind domains
18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following
More informationMath 104B: Number Theory II (Winter 2012)
Math 104B: Number Theory II (Winter 01) Alina Bucur Contents 1 Review 11 Prime numbers 1 Euclidean algorithm 13 Multilicative functions 14 Linear diohantine equations 3 15 Congruences 3 Primes as sums
More informationWhy Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack
Proof Techniques By Chuck Cusack Why Proofs? Writing roofs is not most student s favorite activity. To make matters worse, most students do not understand why it is imortant to rove things. Here are just
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationPOINTS ON CONICS MODULO p
POINTS ON CONICS MODULO TEAM 2: JONGMIN BAEK, ANAND DEOPURKAR, AND KATHERINE REDFIELD Abstract. We comute the number of integer oints on conics modulo, where is an odd rime. We extend our results to conics
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More informationThe Arm Prime Factors Decomposition
The Arm Prime Factors Decomosition Arm Boris Nima arm.boris@gmail.com Abstract We introduce the Arm rime factors decomosition which is the equivalent of the Taylor formula for decomosition of integers
More informationAn Overview of Witt Vectors
An Overview of Witt Vectors Daniel Finkel December 7, 2007 Abstract This aer offers a brief overview of the basics of Witt vectors. As an alication, we summarize work of Bartolo and Falcone to rove that
More informationThe Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001
The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001 The Hasse-Minkowski Theorem rovides a characterization of the rational quadratic forms. What follows is a roof of the Hasse-Minkowski
More informationHENSEL S LEMMA KEITH CONRAD
HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar
More informationp-adic Measures and Bernoulli Numbers
-Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,
More informationDISCRIMINANTS IN TOWERS
DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K/F be a finite searable extension field, and let B be the integral closure of A in K. In this note, we will
More informationANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM
ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationFactorability in the ring Z[ 5]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Paers in Mathematics Mathematics, Deartment of 4-2004 Factorability in the ring
More informationPHYS 301 HOMEWORK #9-- SOLUTIONS
PHYS 0 HOMEWORK #9-- SOLUTIONS. We are asked to use Dirichlet' s theorem to determine the value of f (x) as defined below at x = 0, ± /, ± f(x) = 0, - < x
More informationMA3H1 TOPICS IN NUMBER THEORY PART III
MA3H1 TOPICS IN NUMBER THEORY PART III SAMIR SIKSEK 1. Congruences Modulo m In quadratic recirocity we studied congruences of the form x 2 a (mod ). We now turn our attention to situations where is relaced
More informationIntrinsic Approximation on Cantor-like Sets, a Problem of Mahler
Intrinsic Aroximation on Cantor-like Sets, a Problem of Mahler Ryan Broderick, Lior Fishman, Asaf Reich and Barak Weiss July 200 Abstract In 984, Kurt Mahler osed the following fundamental question: How
More informationPractice Final Solutions
Practice Final Solutions 1. True or false: (a) If a is a sum of three squares, and b is a sum of three squares, then so is ab. False: Consider a 14, b 2. (b) No number of the form 4 m (8n + 7) can be written
More informationMobius Functions, Legendre Symbols, and Discriminants
Mobius Functions, Legendre Symbols, and Discriminants 1 Introduction Zev Chonoles, Erick Knight, Tim Kunisky Over the integers, there are two key number-theoretic functions that take on values of 1, 1,
More informationf(r) = a d n) d + + a0 = 0
Math 400-00/Foundations of Algebra/Fall 07 Polynomials at the Foundations: Roots Next, we turn to the notion of a root of a olynomial in Q[x]. Definition 8.. r Q is a rational root of fx) Q[x] if fr) 0.
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCIT POOJA PATEL Abstract. This aer is an self-contained exosition of the law of uadratic recirocity. We will give two roofs of the Chinese remainder theorem and a roof of uadratic recirocity.
More informationCharacteristics of Fibonacci-type Sequences
Characteristics of Fibonacci-tye Sequences Yarden Blausa May 018 Abstract This aer resents an exloration of the Fibonacci sequence, as well as multi-nacci sequences and the Lucas sequence. We comare and
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More informationSection 0.10: Complex Numbers from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative
Section 0.0: Comlex Numbers from Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationOn the Rank of the Elliptic Curve y 2 = x(x p)(x 2)
On the Rank of the Ellitic Curve y = x(x )(x ) Jeffrey Hatley Aril 9, 009 Abstract An ellitic curve E defined over Q is an algebraic variety which forms a finitely generated abelian grou, and the structure
More informationBrownian Motion and Random Prime Factorization
Brownian Motion and Random Prime Factorization Kendrick Tang June 4, 202 Contents Introduction 2 2 Brownian Motion 2 2. Develoing Brownian Motion.................... 2 2.. Measure Saces and Borel Sigma-Algebras.........
More informationDIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS. 1. Introduction
DIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS INNA ZAKHAREVICH. Introduction It is a well-known fact that there are infinitely many rimes. However, it is less clear how the rimes are distributed
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6.2 Introduction We extend the concet of a finite series, met in Section 6., to the situation in which the number of terms increase without bound. We define what is meant by an infinite
More informationElliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015
18.783 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to
More informationDIVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
IVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUARATIC FIELS PAUL JENKINS AN KEN ONO Abstract. In a recent aer, Guerzhoy obtained formulas for certain class numbers as -adic limits of traces of singular
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationChapter 7 Rational and Irrational Numbers
Chater 7 Rational and Irrational Numbers In this chater we first review the real line model for numbers, as discussed in Chater 2 of seventh grade, by recalling how the integers and then the rational numbers
More informationOn Wald-Type Optimal Stopping for Brownian Motion
J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationCryptanalysis of Pseudorandom Generators
CSE 206A: Lattice Algorithms and Alications Fall 2017 Crytanalysis of Pseudorandom Generators Instructor: Daniele Micciancio UCSD CSE As a motivating alication for the study of lattice in crytograhy we
More informationarxiv: v1 [math.nt] 4 Nov 2015
Wall s Conjecture and the ABC Conjecture George Grell, Wayne Peng August 0, 018 arxiv:1511.0110v1 [math.nt] 4 Nov 015 Abstract We show that the abc conjecture of Masser-Oesterlé-Sziro for number fields
More informationPretest (Optional) Use as an additional pacing tool to guide instruction. August 21
Trimester 1 Pretest (Otional) Use as an additional acing tool to guide instruction. August 21 Beyond the Basic Facts In Trimester 1, Grade 8 focus on multilication. Daily Unit 1: Rational vs. Irrational
More informationSeries Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form.
Series Handout A. Determine which of the following sums are geometric. If the sum is geometric, exress the sum in closed form. 70 a) k= ( k ) b) 50 k= ( k )2 c) 60 k= ( k )k d) 60 k= (.0)k/3 2. Find the
More informationARITHMETIC PROGRESSIONS OF POLYGONAL NUMBERS WITH COMMON DIFFERENCE A POLYGONAL NUMBER
#A43 INTEGERS 17 (2017) ARITHMETIC PROGRESSIONS OF POLYGONAL NUMBERS WITH COMMON DIFFERENCE A POLYGONAL NUMBER Lenny Jones Deartment of Mathematics, Shiensburg University, Shiensburg, Pennsylvania lkjone@shi.edu
More information0.6 Factoring 73. As always, the reader is encouraged to multiply out (3
0.6 Factoring 7 5. The G.C.F. of the terms in 81 16t is just 1 so there is nothing of substance to factor out from both terms. With just a difference of two terms, we are limited to fitting this olynomial
More informationTowards understanding the Lorenz curve using the Uniform distribution. Chris J. Stephens. Newcastle City Council, Newcastle upon Tyne, UK
Towards understanding the Lorenz curve using the Uniform distribution Chris J. Stehens Newcastle City Council, Newcastle uon Tyne, UK (For the Gini-Lorenz Conference, University of Siena, Italy, May 2005)
More informationSOME SUMS OVER IRREDUCIBLE POLYNOMIALS
SOME SUMS OVER IRREDUCIBLE POLYNOMIALS DAVID E SPEYER Abstract We rove a number of conjectures due to Dinesh Thakur concerning sums of the form P hp ) where the sum is over monic irreducible olynomials
More informationMAS 4203 Number Theory. M. Yotov
MAS 4203 Number Theory M. Yotov June 15, 2017 These Notes were comiled by the author with the intent to be used by his students as a main text for the course MAS 4203 Number Theory taught at the Deartment
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More information16.2. Infinite Series. Introduction. Prerequisites. Learning Outcomes
Infinite Series 6. Introduction We extend the concet of a finite series, met in section, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series
More informationApplicable Analysis and Discrete Mathematics available online at HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS
Alicable Analysis and Discrete Mathematics available online at htt://efmath.etf.rs Al. Anal. Discrete Math. 4 (010), 3 44. doi:10.98/aadm1000009m HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS Zerzaihi
More informationPositive decomposition of transfer functions with multiple poles
Positive decomosition of transfer functions with multile oles Béla Nagy 1, Máté Matolcsi 2, and Márta Szilvási 1 Deartment of Analysis, Technical University of Budaest (BME), H-1111, Budaest, Egry J. u.
More informationPractice Final Solutions
Practice Final Solutions 1. Find integers x and y such that 13x + 1y 1 SOLUTION: By the Euclidean algorithm: One can work backwards to obtain 1 1 13 + 2 13 6 2 + 1 1 13 6 2 13 6 (1 1 13) 7 13 6 1 Hence
More informationThe Group of Primitive Almost Pythagorean Triples
The Grou of Primitive Almost Pythagorean Triles Nikolai A. Krylov and Lindsay M. Kulzer arxiv:1107.2860v2 [math.nt] 9 May 2012 Abstract We consider the triles of integer numbers that are solutions of the
More informationMATH 361: NUMBER THEORY ELEVENTH LECTURE
MATH 361: NUMBER THEORY ELEVENTH LECTURE The subjects of this lecture are characters, Gauss sums, Jacobi sums, and counting formulas for olynomial equations over finite fields. 1. Definitions, Basic Proerties
More informationMATH342 Practice Exam
MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice
More informationReal Analysis 1 Fall Homework 3. a n.
eal Analysis Fall 06 Homework 3. Let and consider the measure sace N, P, µ, where µ is counting measure. That is, if N, then µ equals the number of elements in if is finite; µ = otherwise. One usually
More informationGOOD MODELS FOR CUBIC SURFACES. 1. Introduction
GOOD MODELS FOR CUBIC SURFACES ANDREAS-STEPHAN ELSENHANS Abstract. This article describes an algorithm for finding a model of a hyersurface with small coefficients. It is shown that the aroach works in
More informationJohn Weatherwax. Analysis of Parallel Depth First Search Algorithms
Sulementary Discussions and Solutions to Selected Problems in: Introduction to Parallel Comuting by Viin Kumar, Ananth Grama, Anshul Guta, & George Karyis John Weatherwax Chater 8 Analysis of Parallel
More informationWe collect some results that might be covered in a first course in algebraic number theory.
1 Aendices We collect some results that might be covered in a first course in algebraic number theory. A. uadratic Recirocity Via Gauss Sums A1. Introduction In this aendix, is an odd rime unless otherwise
More informationInfinitely Many Quadratic Diophantine Equations Solvable Everywhere Locally, But Not Solvable Globally
Infinitely Many Quadratic Diohantine Equations Solvable Everywhere Locally, But Not Solvable Globally R.A. Mollin Abstract We resent an infinite class of integers 2c, which turn out to be Richaud-Degert
More informationSUMS OF TWO SQUARES PAIR CORRELATION & DISTRIBUTION IN SHORT INTERVALS
SUMS OF TWO SQUARES PAIR CORRELATION & DISTRIBUTION IN SHORT INTERVALS YOTAM SMILANSKY Abstract. In this work we show that based on a conjecture for the air correlation of integers reresentable as sums
More informationCMSC 425: Lecture 4 Geometry and Geometric Programming
CMSC 425: Lecture 4 Geometry and Geometric Programming Geometry for Game Programming and Grahics: For the next few lectures, we will discuss some of the basic elements of geometry. There are many areas
More informationMATH 248A. THE CHARACTER GROUP OF Q. 1. Introduction
MATH 248A. THE CHARACTER GROUP OF Q KEITH CONRAD 1. Introduction The characters of a finite abelian grou G are the homomorhisms from G to the unit circle S 1 = {z C : z = 1}. Two characters can be multilied
More informationTHE THEORY OF NUMBERS IN DEDEKIND RINGS
THE THEORY OF NUMBERS IN DEDEKIND RINGS JOHN KOPPER Abstract. This aer exlores some foundational results of algebraic number theory. We focus on Dedekind rings and unique factorization of rime ideals,
More informationCDH/DDH-Based Encryption. K&L Sections , 11.4.
CDH/DDH-Based Encrytion K&L Sections 8.3.1-8.3.3, 11.4. 1 Cyclic grous A finite grou G of order q is cyclic if it has an element g of q. { 0 1 2 q 1} In this case, G = g = g, g, g,, g ; G is said to be
More informationRAMANUJAN-NAGELL CUBICS
RAMANUJAN-NAGELL CUBICS MARK BAUER AND MICHAEL A. BENNETT ABSTRACT. A well-nown result of Beuers [3] on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity x 2 2
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationRound-off Errors and Computer Arithmetic - (1.2)
Round-off Errors and Comuter Arithmetic - (.). Round-off Errors: Round-off errors is roduced when a calculator or comuter is used to erform real number calculations. That is because the arithmetic erformed
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013 In this lecture we lay the groundwork needed to rove the Hasse-Minkowski theorem for Q, which states that a quadratic form over
More information01. Simplest example phenomena
(March, 20) 0. Simlest examle henomena Paul Garrett garrett@math.umn.edu htt://www.math.umn.edu/ garrett/ There are three tyes of objects in lay: rimitive/rimordial (integers, rimes, lattice oints,...)
More informationNumber Theory Naoki Sato
Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO
More informationTRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES
Khayyam J. Math. DOI:10.22034/kjm.2019.84207 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and
More informationMultiplicative group law on the folium of Descartes
Multilicative grou law on the folium of Descartes Steluţa Pricoie and Constantin Udrişte Abstract. The folium of Descartes is still studied and understood today. Not only did it rovide for the roof of
More informationThe inverse Goldbach problem
1 The inverse Goldbach roblem by Christian Elsholtz Submission Setember 7, 2000 (this version includes galley corrections). Aeared in Mathematika 2001. Abstract We imrove the uer and lower bounds of the
More informationThe Arm Prime Factors Decomposition
The Arm Prime Factors Decomosition Boris Arm To cite this version: Boris Arm. The Arm Prime Factors Decomosition. 2013. HAL Id: hal-00810545 htts://hal.archives-ouvertes.fr/hal-00810545 Submitted on 10
More informationMATH 210A, FALL 2017 HW 5 SOLUTIONS WRITTEN BY DAN DORE
MATH 20A, FALL 207 HW 5 SOLUTIONS WRITTEN BY DAN DORE (If you find any errors, lease email ddore@stanford.edu) Question. Let R = Z[t]/(t 2 ). Regard Z as an R-module by letting t act by the identity. Comute
More informationON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX)0000-0 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS
More informationTHE CHARACTER GROUP OF Q
THE CHARACTER GROUP OF Q KEITH CONRAD 1. Introduction The characters of a finite abelian grou G are the homomorhisms from G to the unit circle S 1 = {z C : z = 1}. Two characters can be multilied ointwise
More informationA Curious Property of the Decimal Expansion of Reciprocals of Primes
A Curious Proerty of the Decimal Exansion of Recirocals of Primes Amitabha Triathi January 6, 205 Abstract For rime 2, 5, the decimal exansion of / is urely eriodic. For those rime for which the length
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationIntroduction to Group Theory Note 1
Introduction to Grou Theory Note July 7, 009 Contents INTRODUCTION. Examles OF Symmetry Grous in Physics................................. ELEMENT OF GROUP THEORY. De nition of Grou................................................
More informationDiophantine Equations and Congruences
International Journal of Algebra, Vol. 1, 2007, no. 6, 293-302 Diohantine Equations and Congruences R. A. Mollin Deartment of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada,
More informationOn the Toppling of a Sand Pile
Discrete Mathematics and Theoretical Comuter Science Proceedings AA (DM-CCG), 2001, 275 286 On the Toling of a Sand Pile Jean-Christohe Novelli 1 and Dominique Rossin 2 1 CNRS, LIFL, Bâtiment M3, Université
More informationFrobenius Elements, the Chebotarev Density Theorem, and Reciprocity
Frobenius Elements, the Chebotarev Density Theorem, and Recirocity Dylan Yott July 30, 204 Motivation Recall Dirichlet s theorem from elementary number theory. Theorem.. For a, m) =, there are infinitely
More informationGAUSSIAN INTEGERS HUNG HO
GAUSSIAN INTEGERS HUNG HO Abstract. We will investigate the ring of Gaussian integers Z[i] = {a + bi a, b Z}. First we will show that this ring shares an imortant roerty with the ring of integers: every
More informationWhen do Fibonacci invertible classes modulo M form a subgroup?
Calhoun: The NPS Institutional Archive DSace Reository Faculty and Researchers Faculty and Researchers Collection 2013 When do Fibonacci invertible classes modulo M form a subgrou? Luca, Florian Annales
More informationComplex Analysis Homework 1
Comlex Analysis Homework 1 Steve Clanton Sarah Crimi January 27, 2009 Problem Claim. If two integers can be exressed as the sum of two squares, then so can their roduct. Proof. Call the two squares that
More informationQUADRATIC RESIDUES AND DIFFERENCE SETS
QUADRATIC RESIDUES AND DIFFERENCE SETS VSEVOLOD F. LEV AND JACK SONN Abstract. It has been conjectured by Sárközy that with finitely many excetions, the set of quadratic residues modulo a rime cannot be
More informationCOMMUNICATION BETWEEN SHAREHOLDERS 1
COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov
More informationA review of the foundations of perfectoid spaces
A review of the foundations of erfectoid saces (Notes for some talks in the Fargues Fontaine curve study grou at Harvard, Oct./Nov. 2017) Matthew Morrow Abstract We give a reasonably detailed overview
More informationAveraging sums of powers of integers and Faulhaber polynomials
Annales Mathematicae et Informaticae 42 (20. 0 htt://ami.ektf.hu Averaging sums of owers of integers and Faulhaber olynomials José Luis Cereceda a a Distrito Telefónica Madrid Sain jl.cereceda@movistar.es
More informationMath 5330 Spring Notes Prime Numbers
Math 5330 Sring 208 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
More information