CS 103X: Discrete Structures Homework Assignment 2 — Solutions
... number p/q, we would have √6 = (p − 5q)/2q, which is a rational number. But we’ve proved the first part that 6 is irrational, which proves the result √ by contradiction. √ in √ For 2 + 6, the same method works, only we use the fact that 3 is irrational. Exercise 8 (20 points). Consider n lines in th ...
... number p/q, we would have √6 = (p − 5q)/2q, which is a rational number. But we’ve proved the first part that 6 is irrational, which proves the result √ by contradiction. √ in √ For 2 + 6, the same method works, only we use the fact that 3 is irrational. Exercise 8 (20 points). Consider n lines in th ...
Chapter 1 Introduction to prime number theory
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
on the nonexistence of odd perfect numbers
... latest, found on June 1, 1999 by Nayan Hajratwala, was part of the Great Internet Mersenne Prime Search (GIMPS) (see http://www.mersenne.org/); it is ...
... latest, found on June 1, 1999 by Nayan Hajratwala, was part of the Great Internet Mersenne Prime Search (GIMPS) (see http://www.mersenne.org/); it is ...
Elementary sieve methods and Brun`s theorem on twin primes
... that there exist infinitely many prime numbers. However, this is not the case for twin primes. We don’t know if there exist finite or infinitely many twin primes, but the Norwegian mathematician Viggo Brun proved that the sum of the reciprocals of all twin primes p is convergent, and converges to Br ...
... that there exist infinitely many prime numbers. However, this is not the case for twin primes. We don’t know if there exist finite or infinitely many twin primes, but the Norwegian mathematician Viggo Brun proved that the sum of the reciprocals of all twin primes p is convergent, and converges to Br ...
Sample Exam 2 Math 221H Fall 2008 Name: Score
... for the associated homogeneous problem y + 8y = 0 is 0 = r2 + 8r = r (r + 8), so the general solution for the homogeneous equation is yc (x) = c1 1 + c2 e−8x . Also, the family of derivatives of f (x) = −25 is generated by the single function g (x) = 1, but this is a solution to the homogeneous equa ...
... for the associated homogeneous problem y + 8y = 0 is 0 = r2 + 8r = r (r + 8), so the general solution for the homogeneous equation is yc (x) = c1 1 + c2 e−8x . Also, the family of derivatives of f (x) = −25 is generated by the single function g (x) = 1, but this is a solution to the homogeneous equa ...
29 APPROXIMATION EXPONENTS FOR FUNCTION
... of a polynomial, we can use the same definition as above for the exponent in this situation. Mahler proved analogues of Dirichlet and Liouville bounds by essentially the same proofs. An analogue of Khintchine’s theorem giving the behaviour of “almost all” functions can also be proved similarly. D. Fe ...
... of a polynomial, we can use the same definition as above for the exponent in this situation. Mahler proved analogues of Dirichlet and Liouville bounds by essentially the same proofs. An analogue of Khintchine’s theorem giving the behaviour of “almost all” functions can also be proved similarly. D. Fe ...
Methods of Proof
... the “arbitrariness” of m and n, we must allow, at the least, that they be different. We accomplish this by choosing different letters a and b in our representations of m and n as “twice an integer plus one.” There is nothing sacred about a and b; we could have used k and `, or x and y, or α and β, o ...
... the “arbitrariness” of m and n, we must allow, at the least, that they be different. We accomplish this by choosing different letters a and b in our representations of m and n as “twice an integer plus one.” There is nothing sacred about a and b; we could have used k and `, or x and y, or α and β, o ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".