number theory and combinatorics

Ngày tải lên: 31/03/2014, 16:22

Ngày tải lên: 31/03/2014, 16:22

... applications of number theory have also broadened In addition to elementary and analytic number theory, increasing use has been made of algebraic number theory (primality testing with Gauss and Jacobi ... Lectures on Elementary Number Theory, Krieger, 1977 K H Rosen, Elementary Number Theory and its Applications, 3rd ed., Addison-Wesley, 1993 M R Schroeder, Number Theory in Science and Communication, ... Problems in Number Theory, Springer-Verlag, 1982 G H Hardy and E M Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, 1979 W J LeVeque, Ftrndamentals of Number Theory, ...

Ngày tải lên: 31/03/2014, 16:20

... of m and n On the other hand, suppose e is a divisor of m and n: e | m, n Then, working downwards, we find successively that e | m and e | n =⇒ e | r1 , e | r1 and e | m =⇒ e | r2 , e | r2 and ... exists a unique number d ∈ N such that d | m, d | n, and furthermore, if e ∈ N then e | m, e | n =⇒ e | d Definition 1.4 We call this number d the greatest common divisor of m and n, and we write ... bottom, d = rt | rt−1 , d | rt and d | rt−1 =⇒ d | rt−2 , d | rt−1 and d | rt−2 =⇒ d | rt−3 , d | r3 and d | r2 =⇒ d | r1 , d | r2 and d | r1 =⇒ d | m, d | r1 and d | m =⇒ d | n Thus d | m,...

Ngày tải lên: 31/03/2014, 16:21

... Generating a random number from a given interval 9.3 The generate and test paradigm 9.4 Generating a random prime 9.5 Generating a random non-increasing sequence 9.6 Generating a random factored number ... probability and independence 8.3 Random variables 8.4 Expectation and variance 8.5 Some useful bounds 8.6 Balls and bins 8.7 Hash functions 8.8 Statistical distance 8.9 Measures of randomness and the ... have (i) a | a, | a, and a | 0; (ii) | a if and only if a = 0; (iii) a | b if and only if −a | b if and only if a | −b; (iv) a | b and a | c implies a | (b + c); (v) a | b and b | c implies a...

Ngày tải lên: 03/04/2014, 12:23

... Catalan Numbers And Hankel Determinants For an integer p ≥ we shall denote the numbers pm+1 pm+1 as generalized Catalan m numbers The Catalan numbers occur for p = (The notion “generalized Catalan numbers” ... is shown that Cm are the Motzkin numbers, Cm are the Catalan numbers (3,3) and Cm are restricted hexagonal numbers Guy [41] gave an interpretation of the (4,4) numbers Cm starting with 1, 4, 17, ... identities References [1] M Aigner, “Motzkin numbers”, Europ J Combinatorics 19, 1998, 663 – 675 [2] M Aigner, “Catalan-like numbers and determinants”, J Combin Theory Ser A 87, 1999, 33 – 51 [3] M Aigner,...

Ngày tải lên: 07/08/2014, 06:22

... Systems, Number theory, and Random matrices, with lectures by E Bogomolny on Quantum and arithmetical chaos, J Conrey on L-functions and random matrix theory, J.-C Yoccoz on Interval exchange maps, and ... physicists and mathematicians, and was the occasion of long and passionate discussions The seminars were published in a book entitled Number Theory and Physics”, J.-M Luck, P Moussa, and M Waldschmidt ... Part I Random matrices: from Physics to Number theory Quantum and Arithmetical Chaos Eugene Bogomolny Notes on L-functions and Random Matrix Theory...

Ngày tải lên: 12/02/2014, 16:20

... Systems, Number theory, and Random matrices, with lectures by E Bogomolny on Quantum and arithmetical chaos, J Conrey on L-functions and random matrix theory, J.-C Yoccoz on Interval exchange maps, and ... physicists and mathematicians, and was the occasion of long and passionate discussions The seminars were published in a book entitled Number Theory and Physics”, J.-M Luck, P Moussa, and M Waldschmidt ... Frontiers in Number Theory, Physics, and Geometry I Pierre Cartier Bernard Julia Pierre Moussa Pierre Vanhove (Eds.) Frontiers in Number Theory, Physics, and Geometry I On Random Matrices,...

Ngày tải lên: 12/02/2014, 17:20

... Theorem Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated by algorithms and explicit computation Active ... let N = {1, 2, 3, } denote the natural numbers, and use the standard notation Z, Q, R, and C for the rings of integer, rational, real, and complex numbers, respectively In this book, we will ... so q = and r = 986 Notice that if a natural number d divides both 2261 and 1275, then d divides their difference 986 and d still divides 1275 On the other hand, if d divides both 1275 and 986,...

Ngày tải lên: 07/03/2014, 16:20

... And likewise if al is odd and between and p - 1, so is az And similarly with al’ and a,’ 46 Solved and Unsolved Problems in Number Theory From Perfect Numbers to the Quadratic Rcciprocity L ... follows: "If a number A divides the difference of two numbers B and C, B and C are called congruent with respect to A , and if not, incongruent A is called the modulus; each of the numbers B and C are ... to expand the fraction a/D into a continued fraction 12 Solved and Unsolved Problems in Number Theory From Perfect Numbers to the Quadratic Reciprocity L a w Thus 13 Any even perfect number...

Ngày tải lên: 15/03/2014, 16:03

... Countable sets Power sets and their cardinality The real numbers are uncountable Problem Set 53 53 55 55 57 59 60 Index 61 CHAPTER Basic Number Theory The natural numbers The natural numbers 0, 1, 2, ... gcd(190, 72) and proceeding as follows Example 1.8 Find the greatest common divisor of 190 and 72 and express it as an integral linear combination of these numbers 6 BASIC NUMBER THEORY Solution ... is the key ingredient 10 BASIC NUMBER THEORY Lemma 1.11 Suppose that a, b ∈ N0 are coprime and n ∈ Z If a | n and b | n, then ab | n Proof Let a | and b | n and choose r, s ∈ Z so that n = =...

Ngày tải lên: 31/03/2014, 16:21

... Chapter Class numbers and class groups of algebraic groups 8.1 Class numbers of algebraic groups and number of classes in a genus 8.2 Class numbers and class groups ... the number of classes in the genus of quadratic forms and of the class numbers of algebraic number fields influenced the study of class numbers of arbitrary algebraic groups defined over a number ... clearly compact groups, and Ui and Ci are open in UD and D(') respectively (and, moreover, generate a base of neighborhoods of the identity), and the indexes [U : Ui] and [D(') : Ci] are finite...

Ngày tải lên: 31/03/2014, 16:21

... mathematics, Number Theory, is noted for its theoretical depth and applications to other fields, including representation theory, physics, and cryptography The forefront of Number Theory is replete ... 1), and n and n − are consecutive numbers, so they cannot both be squares We thus assume k and p are coprime, in which case k and k − p are coprime Thus k − pk is a square if and only if k and ... an even number is of the form 2m, for some integer m; 3) the sum of two odd numbers is an even number; 4) the sum of two even numbers is an even number; 5) the sum of an odd and even number is...

Ngày tải lên: 01/06/2014, 19:42

... on algebra and algebraic number theory, G Eisenstein (1823–1852), noted for his profound work on number theory and elliptic functions, A Enneper (1830–1885), known for his work on the theory of ... introduced complex numbers, his Gaussian integers, into the realm of number theory ([G.1], pp 169–178, 93–148, 313–385; [R]) This was Gauß’ last long paper on number theory, and a very important ... both hands, and keeps his eyes, when not covered with his hands, mostly shut He uses no notes, inside his hands he sees an imaginary calculation, and reads it out to us — that we understand it...

Ngày tải lên: 27/06/2014, 14:20

... Analytic Number Theory A Tribute to Gauss and Dirichlet Clay Mathematics Proceedings Volume Analytic Number Theory A Tribute to Gauss and Dirichlet William Duke Yuri Tschinkel ... international gathering of leading number theorists who reported on recent advances in both classical analytic number theory as well as in related parts of number theory and algebraic geometry It is ... introduced complex numbers, his Gaussian integers, into the realm of number theory ([G.1], pp 169–178, 93–148, 313–385; [R]) This was Gauß’ last long paper on number theory, and a very important...

Ngày tải lên: 06/08/2014, 01:21

... on algebra and algebraic number theory, G Eisenstein (1823–1852), noted for his profound work on number theory and elliptic functions, A Enneper (1830–1885), known for his work on the theory of ... both hands, and keeps his eyes, when not covered with his hands, mostly shut He uses no notes, inside his hands he sees an imaginary calculation, and reads it out to us — that we understand it ... teacher in Germany to give lectures on his favourite subject, number theory, and on the application of analytical techniques to number theory; 23 of his lectures were devoted to these topics ([Bi.1];...

Ngày tải lên: 06/08/2014, 01:21

... Narkiewicz, W.: Elementary and analytic theory of algebraic numbers Warszawa: PWN – Polish Scientific Publishers, 1974 [N.2] Narkiewicz, W.: The development of prime number theory Berlin etc.: Springer, ... Shields, A.: Lejeune Dirichlet and the birth of analytic number theory: 1837–1839 Math Intell 11, 7–11 (1989) [Si] Siegel, C.L.: Lectures on advanced analytic number theory Bombay: Tata Institute ... started to write a report on the theory of numbers beginning with the investigations of P de Fermat and ending with the then (1865) latest results of Kummer, Kronecker, and Hurwitz The six parts of...

Ngày tải lên: 06/08/2014, 01:21

... P1 and P2 ”, J Number Theory 101 (2003), no 1, p 195–207 T D Browning – “Counting rational points on del Pezzo surfaces of degree five”, in Proceedings of the Session in Analytic Number Theory and ... a natural number with t ≥ 6, and let c1 , , ct be natural numbers satisfying (c1 , , ct ) = Then for each natural number d there is a positive number ∆, depending at most on c and d, with ... real numbers α with α ∈ [0, 1) for which there exist a ∈ Z and q ∈ N satisfying ≤ a ≤ q ≤ X, (a, q) = and |qα − a| ≤ XP −3 We then define sets of major arcs M = M(P 3/4 ) and K = M(Q1/4 ), and...

Ngày tải lên: 06/08/2014, 01:21