Task - Illustrative Mathematics
... as they relate to the notions of rationality and irrationality. As such, this task perhaps makes most sense after students learn the key terms (rational and irrational numbers), as well as examples of each (e.g., the irrationality of √2 ‾, π, etc.), but before formally proving any of the statements ...
... as they relate to the notions of rationality and irrationality. As such, this task perhaps makes most sense after students learn the key terms (rational and irrational numbers), as well as examples of each (e.g., the irrationality of √2 ‾, π, etc.), but before formally proving any of the statements ...
On the least common multiple of q
... if and only if d does not divide n + 1. Proof. Suppose that (5) holds for some positive integer k. Let k≡a ...
... if and only if d does not divide n + 1. Proof. Suppose that (5) holds for some positive integer k. Let k≡a ...
2014 Intermediate Solutions
... However, this does not provide a full mathematical explanation that would be acceptable if you were just given the question without any alternative answers. So usually we have included a complete solution which does not use the fact that one of the given alternatives is correct. Thus we have aimed t ...
... However, this does not provide a full mathematical explanation that would be acceptable if you were just given the question without any alternative answers. So usually we have included a complete solution which does not use the fact that one of the given alternatives is correct. Thus we have aimed t ...
For screen
... n 6≡ 0 (mod 3) and n 6≡ 0 (mod 4). Furthermore, if such solution exists, known results on Catalan’s equation imply that n must be odd. Indeed, if q0 n = 2m, then xm 0 − y1 = −1 for some positive integer y1 dividing y, and this is impossible since x0 < 1011 , by a result of Hyyrö (see [13], pages 26 ...
... n 6≡ 0 (mod 3) and n 6≡ 0 (mod 4). Furthermore, if such solution exists, known results on Catalan’s equation imply that n must be odd. Indeed, if q0 n = 2m, then xm 0 − y1 = −1 for some positive integer y1 dividing y, and this is impossible since x0 < 1011 , by a result of Hyyrö (see [13], pages 26 ...
![arXiv:math/0608068v1 [math.NT] 2 Aug 2006](http://s1.studyres.com/store/data/016444901_1-15a2bfb469428d063b2c2745e8d56343-300x300.png)
Number theory
Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called ""The Queen of Mathematics"" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation).The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ""number theory"". (The word ""arithmetic"" is used by the general public to mean ""elementary calculations""; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.