elementary number theory notes - santos

... Arithmetic-Mean-Geometric-Mean Inequality for n = 2. Assume that the Arithmetic-Mean-Geometric-Mean Inequality holds true for n = 2 k−1 , k > 2, that is, assume that nonnegative real numbers ... 2 n subsets. 33 APS Prove that if n is a natural number, n 5 /5 + n 4 /2 + n 3 /3 − n/30 is always an integer. Elementary Number Theory Notes c  David A. Santos January 15, 2004 14 Cha...

Ngày tải lên: 31/03/2014, 16:21

... [LL93]), which is the best-known general purpose factoriza- tion algorithm. A description of how the number field sieve works is beyond the scope of this book. However, the number field sieve makes ... There are infinitely many composite num- bers. Proof. To obtain a new composite number, multiply together the first n composite numbers and don’t add 1. 12 1. Prime Numbers 1.2.2 Enumerating P...

Ngày tải lên: 07/03/2014, 16:20

Ngày tải lên: 31/03/2014, 16:21

... triangular number t n is the number of dots in a triangular array that has n rows with i dots in the i-th row. Find aformulafort n , n ≥ 1. (b) Suppose that for each n ≥ 1. Let s n be the number ... are per- mitted, provided that all copies and derivatives retain the same permissions. Specifically no commerical use of these notes or any revisions thereof is per- mitted. i 12 CHAPTER 3....

Ngày tải lên: 31/03/2014, 16:21

Ngày tải lên: 31/03/2014, 16:21

... natural numbers N are well-ordered, ie every subset S ⊂ N has a least element. 1.4 The Fundamental Theorem of Arithmetic Proposition 1.4 (Euclid’s Lemma) Suppose p ∈ N is a prime number; and sup- pose ... coprime to n even if n is composite. 5.2 Carmichael numbers Definition 5.2 Suppose n is an odd number > 1. Then we say that n is aCarmichael number if n is not a prime, but is a pseud...

Ngày tải lên: 31/03/2014, 16:21

... Abundant Numbers 260 7.7 Notes 265 8 Prime Numbers 267 8.1 Chebyshev’s Theorems 267 8.2 Mertens’s Theorems 275 8.3 The Number of Prime Divisors of an Integer 282 8.4 Notes 287 9 The Prime Number ... Theorems 486 16.4 Notes 495 References 497 Index 509 Part I A First Course in Number Theory 1 Divisibility and Primes 1.1 Division Algorithm Divisibility is a fundamental concept in...

Ngày tải lên: 31/03/2014, 16:21

... Take a holomorphic theory with field space F hol , and the complex conjugate of the n-point functions. This is a theory of anti-holomorphic fields, with a field space F hol anti-linearly isomorphic ... holomorphic theory with Virasoro field T and G ⊂ F a theory with Virasoro field T 1 , then the complementary sub -theory G  has a Virasoro field T −T 1 . Consider now the holomorphic field...

Ngày tải lên: 12/02/2014, 16:20

... work of Conrey-Ghosh, Conrey-Gonek, Duke-Friedlander-Iwaniec, Kowalski-Michel-Vanderkam, Ju- tila, Motohashi, Ivic, Soundararajan, Rubinstein, and others on moments of families of L-functions which ... conjectures of Keating-Snaith and Conrey-Farmer on moments of zeta- and L-functions and another was the development of the notion of symmetry type of families of L- functions by Katz-Sarnak. Ad...

Ngày tải lên: 12/02/2014, 17:20

... = IU. (3112k - 1) = -1 ’(12k - 113)R = -1 . (-1 13)p = IN (3112k + 5) = (1% + 513)R = (-1 13)p = -IN. (3112k - 5) = -1 (12k - 513)R = -1 .(113)p = -lo. Therefore q ... in number theory . Bibliography: p . Includes index . 1 . Numbers. Theory of . I . Title . [QA241.S44 19781 5E.7 7 7-1 3010 ISBN 0-8 28...

Ngày tải lên: 15/03/2014, 16:03