Rational Numbers - Abstractmath.org
... The integers can be thought of as beads or points in a row going to infinity in both directions. The rationals go to infinity in both directions, too, but: ...
... The integers can be thought of as beads or points in a row going to infinity in both directions. The rationals go to infinity in both directions, too, but: ...
Section 4.3 - The Chinese Remainder Theorem
... (37) Since 1000 ≡ 1 (mod 37), given a number n, starting from the ones digit, break n into chunks of three digits. Then add all these three digit numbers together. The 3-chunk sum is divisible by 37 if and only if n is divisible by 37. (101) Since 100 ≡ −1 (mod 101), given a number n, starting from ...
... (37) Since 1000 ≡ 1 (mod 37), given a number n, starting from the ones digit, break n into chunks of three digits. Then add all these three digit numbers together. The 3-chunk sum is divisible by 37 if and only if n is divisible by 37. (101) Since 100 ≡ −1 (mod 101), given a number n, starting from ...
CHAP05 Distribution of Primes
... Many even numbers have several representations as the sum of two primes. For example 22 is also equal to 5 + 17 and, of course, it is 11 + 11. It might seem that we never need to go beyond 7 for the smaller prime in such a representation but watch when we continue up to 100. ...
... Many even numbers have several representations as the sum of two primes. For example 22 is also equal to 5 + 17 and, of course, it is 11 + 11. It might seem that we never need to go beyond 7 for the smaller prime in such a representation but watch when we continue up to 100. ...
15(3)
... Mathematics Department, University of Santa Clara, Santa Clara, California 95053. All checks ($15.00 per year) should be made out to the Fibonacci Association or The Fibonacci Quarterly. Two copies of manuscripts intended for publication in the Quarterly should be sent to Verner E, Hoggatt, Jr., Mat ...
... Mathematics Department, University of Santa Clara, Santa Clara, California 95053. All checks ($15.00 per year) should be made out to the Fibonacci Association or The Fibonacci Quarterly. Two copies of manuscripts intended for publication in the Quarterly should be sent to Verner E, Hoggatt, Jr., Mat ...
Full text
... that this equation is solvable iff z2 - (k2 - 4)z/2 = +4 is solvable. The latter equation has an obvious solution, namely {z9 y)= (k, 1) . So we have solutions of a2 - kab + b2 = 1 for every k, not just fc = 3. When k = 3, we have only the solutions given by the Theorem of §3, but when k - 4 we have ...
... that this equation is solvable iff z2 - (k2 - 4)z/2 = +4 is solvable. The latter equation has an obvious solution, namely {z9 y)= (k, 1) . So we have solutions of a2 - kab + b2 = 1 for every k, not just fc = 3. When k = 3, we have only the solutions given by the Theorem of §3, but when k - 4 we have ...
File
... Now consider the identity (1 + x) m (1 + x) n ≡ (1 + x) m + n . We shall compare the coefficient of x r (assuming 0 ≤ m ≤ n ≤ r ) on both sides. On the left hand side, we have (1 + x) m (1 + x) n = ( C0m + C1m x + C2m x 2 + ")( C0n + C1n x + C2n x 2 + ") so that the coefficient of x r is C0mCrn + C1 ...
... Now consider the identity (1 + x) m (1 + x) n ≡ (1 + x) m + n . We shall compare the coefficient of x r (assuming 0 ≤ m ≤ n ≤ r ) on both sides. On the left hand side, we have (1 + x) m (1 + x) n = ( C0m + C1m x + C2m x 2 + ")( C0n + C1n x + C2n x 2 + ") so that the coefficient of x r is C0mCrn + C1 ...
Coprime (r,k)-Residue Sets In Z
... Proof. Let A be a mapping from Ik1 (k`) to I (`) defined by A (x) = x mod ` First, we show that Im(A ) ⊆ I (`). Let x ∈ Ik1 (k`). Then, x = a` + b, 0 ≤ b ≤ `. Since x ∈ Ik1 (k`), then by the definition of that set, it follows that x ∈ I (k`). Therefore gcd(x, `) = 1 and consequently gcd(b, `) = 1. T ...
... Proof. Let A be a mapping from Ik1 (k`) to I (`) defined by A (x) = x mod ` First, we show that Im(A ) ⊆ I (`). Let x ∈ Ik1 (k`). Then, x = a` + b, 0 ≤ b ≤ `. Since x ∈ Ik1 (k`), then by the definition of that set, it follows that x ∈ I (k`). Therefore gcd(x, `) = 1 and consequently gcd(b, `) = 1. T ...
Fermat - The Math Forum @ Drexel
... right angled Pythagorean Triple can never be formed for n > 2 and it is this result that confirms Fermat’s Last Theorem. Group 1: When the initial angle Z is obtuse (Z > 90º), Cos Z is negative and z2 > x2 + y2. As we increase n, z2 will always be greater than (x2 + y2) and equality can never exist ...
... right angled Pythagorean Triple can never be formed for n > 2 and it is this result that confirms Fermat’s Last Theorem. Group 1: When the initial angle Z is obtuse (Z > 90º), Cos Z is negative and z2 > x2 + y2. As we increase n, z2 will always be greater than (x2 + y2) and equality can never exist ...
On counting permutations by pairs of congruence classes of major
... formula (A). We now give a bijective proof of (A), which is the special case k = n, ` = n−1 of Theorem 3.1, namely mn (i\n; j\n − 1) = (n − 2)! Proposition 4.1 (Bijection for the case k = n, ` = n − 1 of Theorem 3.1.) Fix integers 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 2. Then the number of permutations σ in Sn ...
... formula (A). We now give a bijective proof of (A), which is the special case k = n, ` = n−1 of Theorem 3.1, namely mn (i\n; j\n − 1) = (n − 2)! Proposition 4.1 (Bijection for the case k = n, ` = n − 1 of Theorem 3.1.) Fix integers 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 2. Then the number of permutations σ in Sn ...
Odd Collatz Sequence and Binary Representations
... We just write OCS if we mean an arbitrary odd Collatz sequence or if the seed is known and in plural form we write OCS’s. Obviously 3n + 1 (i.e. the Collatz conjecture) is solved if we prove that the OCS of any odd number is finite. • The OCS of a number x is cyclic in the same way that a Collatz seq ...
... We just write OCS if we mean an arbitrary odd Collatz sequence or if the seed is known and in plural form we write OCS’s. Obviously 3n + 1 (i.e. the Collatz conjecture) is solved if we prove that the OCS of any odd number is finite. • The OCS of a number x is cyclic in the same way that a Collatz seq ...
Ideal classes and Kronecker bound
... same index, but there are only a finite number of them. We will call C the Kronecker bound since it essentially occurs in Kronecker’s thesis [13, p. 15] in the special case of Q(ζp ) and Kronecker pointed out in another paper later [12, pp. 64–65] that the argument using this bound applies to any nu ...
... same index, but there are only a finite number of them. We will call C the Kronecker bound since it essentially occurs in Kronecker’s thesis [13, p. 15] in the special case of Q(ζp ) and Kronecker pointed out in another paper later [12, pp. 64–65] that the argument using this bound applies to any nu ...
ON THE GENERA OF X0(N) 1. Introduction For each positive integer
... g0 (N ) is even. They are given in the following list, where p denotes a prime and r denotes a positive integer: (1) N = 1, 2, 3, 4, 8 or 16; (2) N = pr where p ≡ 5 (mod 8); (3) N = pr where p ≡ 7 (mod 8) and r odd; (4) N = pr where p ≡ 3 (mod 8) and r even; (5) N = 2pr where p ≡ ±3 (mod 8); (6) N = ...
... g0 (N ) is even. They are given in the following list, where p denotes a prime and r denotes a positive integer: (1) N = 1, 2, 3, 4, 8 or 16; (2) N = pr where p ≡ 5 (mod 8); (3) N = pr where p ≡ 7 (mod 8) and r odd; (4) N = pr where p ≡ 3 (mod 8) and r even; (5) N = 2pr where p ≡ ±3 (mod 8); (6) N = ...
36(2)
... are given followed by more general results. 2.1 Case 1: The Pascal Square and Variations The Pascal array in Table 1 is formed by the use of the recurrence relation ...
... are given followed by more general results. 2.1 Case 1: The Pascal Square and Variations The Pascal array in Table 1 is formed by the use of the recurrence relation ...
Chapter 8.1 – 8.5 - MIT OpenCourseWare
... Number theory is the study of the integers. Why anyone would want to study the integers may not be obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . . . . Which one don’t you understand? What practical value is there in it? The mathematician G. H. ...
... Number theory is the study of the integers. Why anyone would want to study the integers may not be obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . . . . Which one don’t you understand? What practical value is there in it? The mathematician G. H. ...
ON THE NUMBER OF NON-ZERO DIGITS OF INTEGERS IN
... Dedicated to A. Sárközy on the occasion of his 75th birthday Abstract. We prove various finiteness theorems for integers having only few non-zero digits in different multi-base representations simultaneously. ...
... Dedicated to A. Sárközy on the occasion of his 75th birthday Abstract. We prove various finiteness theorems for integers having only few non-zero digits in different multi-base representations simultaneously. ...
Document
... Proof: By strong induction. Let P(n) be “n is the sum of distinct powers of two.” We prove that P(n) is true for all n ∈ ℕ. As our base case, we prove P(0), that 0 is the sum of distinct powers of 2. Since the empty sum of no powers of 2 is equal to 0, P(0) holds. For the inductive step, assume that ...
... Proof: By strong induction. Let P(n) be “n is the sum of distinct powers of two.” We prove that P(n) is true for all n ∈ ℕ. As our base case, we prove P(0), that 0 is the sum of distinct powers of 2. Since the empty sum of no powers of 2 is equal to 0, P(0) holds. For the inductive step, assume that ...
Chapter4p1
... additive inverse of a modulo m and 0 is its own additive inverse. a +m (m− a ) = 0 and 0 +m 0 = 0 Distributivity: If a, b, and c belong to Zm , then a ∙m (b +m c) = (a ∙m b) +m (a ∙m c) and (a +m b) ∙m c = (a ∙m c) +m (b ∙m c). Exercises 42-44 ask for proofs of these properties. Multiplica ...
... additive inverse of a modulo m and 0 is its own additive inverse. a +m (m− a ) = 0 and 0 +m 0 = 0 Distributivity: If a, b, and c belong to Zm , then a ∙m (b +m c) = (a ∙m b) +m (a ∙m c) and (a +m b) ∙m c = (a ∙m c) +m (b ∙m c). Exercises 42-44 ask for proofs of these properties. Multiplica ...
Sample pages 2 PDF
... called axioms. A person who understands decimal addition will clearly be able to answer the following simple Questions: Which of the following equations are definitions? Which ones are theorems? If an equation is a theorem, what is the proof? ...
... called axioms. A person who understands decimal addition will clearly be able to answer the following simple Questions: Which of the following equations are definitions? Which ones are theorems? If an equation is a theorem, what is the proof? ...
Random Number Generator
... Definition. Let a and m be relatively prime positive integers. Then the least positive integer x such that a x = 1 (mod m) is called the order of a modulo m denoted by ord m a. December 1999 ...
... Definition. Let a and m be relatively prime positive integers. Then the least positive integer x such that a x = 1 (mod m) is called the order of a modulo m denoted by ord m a. December 1999 ...
Sample pages 6 PDF
... working with generalized integers, in particular, working in Z[ n] to solve ...
... working with generalized integers, in particular, working in Z[ n] to solve ...
Central Limit Theorem
... is of length 6%, will contain the true value of p. This type of confidence interval is typically reported in the news as follows: this survey has a 3% margin of error. In fact, most of the surveys that one sees reported in the paper will have sample sizes around 1000. A somewhat surprising fact is t ...
... is of length 6%, will contain the true value of p. This type of confidence interval is typically reported in the news as follows: this survey has a 3% margin of error. In fact, most of the surveys that one sees reported in the paper will have sample sizes around 1000. A somewhat surprising fact is t ...
Abelian and non-Abelian numbers via 3D Origami
... We are interested in extending the scope of the referred origami axioms, that is, in adding new axioms to the HJAs in order to get fields larger than O, and studying the arithmetic properties of those new numbers. 1.2. Beyond the Huzita-Justin axioms: n-fold axioms. The description of the folding mo ...
... We are interested in extending the scope of the referred origami axioms, that is, in adding new axioms to the HJAs in order to get fields larger than O, and studying the arithmetic properties of those new numbers. 1.2. Beyond the Huzita-Justin axioms: n-fold axioms. The description of the folding mo ...
In this lecture we will start with Number Theory. We will start
... which states something stronger than this, but we will start small. We formulate this as a lemma: Lemma: Given any natural number n > 1, there exists a prime p such that p|n. Proof: This proof is by contradiction so we assume that there are natural numbers that do not have prime divisors, call them ...
... which states something stronger than this, but we will start small. We formulate this as a lemma: Lemma: Given any natural number n > 1, there exists a prime p such that p|n. Proof: This proof is by contradiction so we assume that there are natural numbers that do not have prime divisors, call them ...