ON THE SET OF POSITIVE INTEGERS WHICH ARE
... Motivated by these two examples, we were led to the following lemmas and a consecutiveness theorem. Lemma 1 is a rather famous result from additive number theory. Lemma 1. Let a and b be relatively prime positive integers. Then ax + by = n has a solution in nonnegative integers x and y, if n is lar ...
... Motivated by these two examples, we were led to the following lemmas and a consecutiveness theorem. Lemma 1 is a rather famous result from additive number theory. Lemma 1. Let a and b be relatively prime positive integers. Then ax + by = n has a solution in nonnegative integers x and y, if n is lar ...
CMSC 203 / 0202 Fall 2002
... More time in class for students to try to solve problems that we then work through as a class In return: Students agree to review chapter readings before class so we can spend less time on the basics without losing everybody. September1999 October 1999 ...
... More time in class for students to try to solve problems that we then work through as a class In return: Students agree to review chapter readings before class so we can spend less time on the basics without losing everybody. September1999 October 1999 ...
Proof of Euler`s φ (Phi) Function Formula - Rose
... Definition 1. For n ≥ 1, φ(n) denotes the number of positive integers not exceeding n and relatively prime to n. For the purpose of self-containment and clarity the gcd is defined as follows. Definition 2 (Greatest Common Divisor). The greatest common divisor of two nonzero integers a and b is the l ...
... Definition 1. For n ≥ 1, φ(n) denotes the number of positive integers not exceeding n and relatively prime to n. For the purpose of self-containment and clarity the gcd is defined as follows. Definition 2 (Greatest Common Divisor). The greatest common divisor of two nonzero integers a and b is the l ...
as a PDF
... Definition 1. For n ≥ 1, φ(n) denotes the number of positive integers not exceeding n and relatively prime to n. For the purpose of self-containment and clarity the gcd is defined as follows. Definition 2 (Greatest Common Divisor). The greatest common divisor of two nonzero integers a and b is the l ...
... Definition 1. For n ≥ 1, φ(n) denotes the number of positive integers not exceeding n and relatively prime to n. For the purpose of self-containment and clarity the gcd is defined as follows. Definition 2 (Greatest Common Divisor). The greatest common divisor of two nonzero integers a and b is the l ...
UbD (Understanding by Design) Lesson Plan
... c. Use representations to model and interpret physical, social, and mathematical phenomena. ...
... c. Use representations to model and interpret physical, social, and mathematical phenomena. ...
Transcendence of generalized Euler constants,
... JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ...
... JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ...
The mystery of the number 1089 – how
... The aim of the present note is to investigate what happens if one replaces three digit numbers by numbers of arbitrary length. More precisely we fix an n ≥ 2, and we will consider n-digit numbers a = a1 a2 . . . an with ai ∈ {0, . . . , 9} and a1 > an . Then we calculate a1 . . . an − an . . . a1 , ...
... The aim of the present note is to investigate what happens if one replaces three digit numbers by numbers of arbitrary length. More precisely we fix an n ≥ 2, and we will consider n-digit numbers a = a1 a2 . . . an with ai ∈ {0, . . . , 9} and a1 > an . Then we calculate a1 . . . an − an . . . a1 , ...
![B2[∞]-sequences of square numbers](http://s1.studyres.com/store/data/014554791_1-cdd0f3de1fbdb07bf0c0f87a9c563b95-300x300.png)
B2[∞]-sequences of square numbers
... Definition. We say that a sequence of positive integers {nk } has the property B2 [∞], or that it is a B2 [∞]-sequence, if P ...
... Definition. We say that a sequence of positive integers {nk } has the property B2 [∞], or that it is a B2 [∞]-sequence, if P ...
Integers, Prime Factorization, and More on Primes
... Proof. (1) By the Euclidean algorithm, there exist integers m, n such that ma + nb = 1. Multiplying c to both sides we have mac + nbc = c. Since a | bc, i.e., bc = qa for some integer q, then c = mac + nqa = (mc + nq)a, which means that a is a divisor of c. (2) If p - a, then gcd(p, a) = 1. Thus by ...
... Proof. (1) By the Euclidean algorithm, there exist integers m, n such that ma + nb = 1. Multiplying c to both sides we have mac + nbc = c. Since a | bc, i.e., bc = qa for some integer q, then c = mac + nqa = (mc + nq)a, which means that a is a divisor of c. (2) If p - a, then gcd(p, a) = 1. Thus by ...
Finite Model Theory
... infinite structures and most of model theory is based on methods that take infiniteness of structures for granted. In that context finite models are anomalies that deserve only marginal attention. Finite model theory arose as an independent field of logic from consideration of problems in theoretica ...
... infinite structures and most of model theory is based on methods that take infiniteness of structures for granted. In that context finite models are anomalies that deserve only marginal attention. Finite model theory arose as an independent field of logic from consideration of problems in theoretica ...
Transcendental values of the digamma function
... M. Ram Murty, N. Saradha / Journal of Number Theory 125 (2007) 298–318 ...
... M. Ram Murty, N. Saradha / Journal of Number Theory 125 (2007) 298–318 ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.