1. Give complete and precise definitions for the following. (a) F is a
... Q(eπi/180 ) is closed under addition and division, so the fact that e−πi/180 = 1/eπi/180 implies cos(1◦ ) ∈ Q(eπi/180 ), completing the proof. Alternatively, we may use a constructablity argument. First, use trigonometric identities to write ...
... Q(eπi/180 ) is closed under addition and division, so the fact that e−πi/180 = 1/eπi/180 implies cos(1◦ ) ∈ Q(eπi/180 ), completing the proof. Alternatively, we may use a constructablity argument. First, use trigonometric identities to write ...
1-1 - cloudfront.net
... Objectives: 1) To use inductive reasoning to make conjectures. 2) To find counterexamples to disprove statements that are false. ...
... Objectives: 1) To use inductive reasoning to make conjectures. 2) To find counterexamples to disprove statements that are false. ...
Primes and Greatest Common Divisors
... There are infinitely many primes Lemma Every natural number greater than one is either prime or it has a prime divisor Follows from fundamental theorem Proof Suppose towards a contradiction that there are only finitely many primes p1 , p2 , p3 , . . . , pk . Consider the number q = p1 p2 p3 . . . p ...
... There are infinitely many primes Lemma Every natural number greater than one is either prime or it has a prime divisor Follows from fundamental theorem Proof Suppose towards a contradiction that there are only finitely many primes p1 , p2 , p3 , . . . , pk . Consider the number q = p1 p2 p3 . . . p ...
21(2)
... where, as usual, [#] denotes the greatest integer in x. The proofs of the above formulas are based on the following lemma. ...
... where, as usual, [#] denotes the greatest integer in x. The proofs of the above formulas are based on the following lemma. ...
Interpolated Schur multiple zeta values
... To include the parameter t into the Graph we will construct a Lattice Graph with two vertices on each position (x, y) together with five different types of edges. Definition 4.1. We define for a fixed N ≥ 1 the weighted directed Graph GtN = (VN , EN , w), which we call t-Lattice Graph, by the follow ...
... To include the parameter t into the Graph we will construct a Lattice Graph with two vertices on each position (x, y) together with five different types of edges. Definition 4.1. We define for a fixed N ≥ 1 the weighted directed Graph GtN = (VN , EN , w), which we call t-Lattice Graph, by the follow ...
Introduction to Number Theory
... facts that you will be able to easily verify: a. If a|b, then a|kb, ka|kb. b. If ka|kb and k 6= 0, then a|b. c. If a|b and b|c, then a|c. d. If a|b and a|c, then a|(mb + nc). e. If a|b and b|a, then a = ±b. f. If a|b, a, b > 0, then a ≥ b. g. For any choice of a, b, there exists a unique q, r, 0 ≤ r ...
... facts that you will be able to easily verify: a. If a|b, then a|kb, ka|kb. b. If ka|kb and k 6= 0, then a|b. c. If a|b and b|c, then a|c. d. If a|b and a|c, then a|(mb + nc). e. If a|b and b|a, then a = ±b. f. If a|b, a, b > 0, then a ≥ b. g. For any choice of a, b, there exists a unique q, r, 0 ≤ r ...
Full text
... A composition of n consists of an ordered sequence of positive integers whose sum is n. A palindromic composition (or palindrome) is one for which the sequence reads the same forwards and backwards. We derive results for the number of "+?5 signs, summands, levels (a suxnmand followed, by itself), ri ...
... A composition of n consists of an ordered sequence of positive integers whose sum is n. A palindromic composition (or palindrome) is one for which the sequence reads the same forwards and backwards. We derive results for the number of "+?5 signs, summands, levels (a suxnmand followed, by itself), ri ...
8(4)
... An elementary algorithm is presented ifor finding the greatest common divisor of two numbers. It is trivial to programme and fast, even for large numbers. Only addition is used, and the only storage space needed is enough to hold the two numbers. About three years ago, I discovered an algorithm whic ...
... An elementary algorithm is presented ifor finding the greatest common divisor of two numbers. It is trivial to programme and fast, even for large numbers. Only addition is used, and the only storage space needed is enough to hold the two numbers. About three years ago, I discovered an algorithm whic ...
OPEN DIOPHANTINE PROBLEMS 1. Diophantine Equations 1.1
... Our goal here is not to describe in detail the state of the art regarding these questions (see for instance [La8]). It suffices to say – that a complete answer to question (a) is not yet available. There is no algorithm (not even a conjectural one) to decide whether a curve has a rational point or n ...
... Our goal here is not to describe in detail the state of the art regarding these questions (see for instance [La8]). It suffices to say – that a complete answer to question (a) is not yet available. There is no algorithm (not even a conjectural one) to decide whether a curve has a rational point or n ...
