The largest prime factor of a Mersenne number
... to the perfect number 2n−1 p, and he found four such primes. Presumably Euclid also knew that if 2n − 1 is prime, then so is n prime, and that the converse does not always hold. In the 18th century, Euler showed that Euclid’s formula for perfect numbers gives rise to all even examples. (It is conjec ...
... to the perfect number 2n−1 p, and he found four such primes. Presumably Euclid also knew that if 2n − 1 is prime, then so is n prime, and that the converse does not always hold. In the 18th century, Euler showed that Euclid’s formula for perfect numbers gives rise to all even examples. (It is conjec ...
Number Theory & RSA
... a square-free, or quadratfrei, integer: one divisible by no perfect square, except 1. Examples: 10 is square-free but 18 is not, as it is divisible by 9 = 32. The smallest square-free numbers: 1, 2, ...
... a square-free, or quadratfrei, integer: one divisible by no perfect square, except 1. Examples: 10 is square-free but 18 is not, as it is divisible by 9 = 32. The smallest square-free numbers: 1, 2, ...
Modular Arithmetic
... • You can often prove statements about congruences by reducing them to statements about divisibility. • You can often prove statements about divisibility by reducing them to equations. (a) Suppose a = b (mod n) and c = d (mod n). a = b (mod n) means n | a − b and c = d (mod n) means n | c − d. By pr ...
... • You can often prove statements about congruences by reducing them to statements about divisibility. • You can often prove statements about divisibility by reducing them to equations. (a) Suppose a = b (mod n) and c = d (mod n). a = b (mod n) means n | a − b and c = d (mod n) means n | c − d. By pr ...
LINEAR INDEPENDENCE OF LOGARITHMS OF - IMJ-PRG
... logarithm of an algebraic number by an algebraic number is usually not a logarithm of an algebraic number. This remark, which goes back to Euler, is the root of our subject. The main (if not the more general) statement of these lectures is the following: Theorem 1.1 (Baker). — If `1 , . . . , `m are ...
... logarithm of an algebraic number by an algebraic number is usually not a logarithm of an algebraic number. This remark, which goes back to Euler, is the root of our subject. The main (if not the more general) statement of these lectures is the following: Theorem 1.1 (Baker). — If `1 , . . . , `m are ...
ON THE ERROR TERM OF THE LOGARITHM OF THE LCM OF A
... where the constant Bf is explicit. The author also proves that for reducible polynomials of degree two, the asymptotic is linear in n. For polynomials of higher degree nothing is known, except for products of linear polynomials, which are studied in [5]. An important ingredient in Cilleruelo’s argum ...
... where the constant Bf is explicit. The author also proves that for reducible polynomials of degree two, the asymptotic is linear in n. For polynomials of higher degree nothing is known, except for products of linear polynomials, which are studied in [5]. An important ingredient in Cilleruelo’s argum ...
The Goldston-Pintz-Yıldırım sieve and some applications
... conjecture. The twin prime conjecture asserts that the gap between consecutive primes is ifinitely often as small as it possibly can be, that is, pn+1 − pn = 2 for infinitely many n. Goldston-Pintz-Yıldırım[15] were able to use their method to prove, conditionally, that pn+1 −pn 6 16 for infinitely ...
... conjecture. The twin prime conjecture asserts that the gap between consecutive primes is ifinitely often as small as it possibly can be, that is, pn+1 − pn = 2 for infinitely many n. Goldston-Pintz-Yıldırım[15] were able to use their method to prove, conditionally, that pn+1 −pn 6 16 for infinitely ...
Untitled
... essentials, without being too elementary, too excessively pedagogical, and too full of distractions. Some of the features of this text are the following: (1) Symbolic logic and the use of logical notation are kept to a minimum. We discuss only what is absolutely necessary—as is the case in most adva ...
... essentials, without being too elementary, too excessively pedagogical, and too full of distractions. Some of the features of this text are the following: (1) Symbolic logic and the use of logical notation are kept to a minimum. We discuss only what is absolutely necessary—as is the case in most adva ...
An Introduction to Combinatorics and Graph Theory
... rolled? Two dice? n dice? As stated, this is ambiguous: what do we mean by “outcome”? Suppose we roll two dice, say a red die and a green die. Is “red two, green three” a different outcome than “red three, green two”? If yes, we are counting the number of possible “physical” outcomes, namely 36. If n ...
... rolled? Two dice? n dice? As stated, this is ambiguous: what do we mean by “outcome”? Suppose we roll two dice, say a red die and a green die. Is “red two, green three” a different outcome than “red three, green two”? If yes, we are counting the number of possible “physical” outcomes, namely 36. If n ...
How to Recognize Whether a Natural Number is a Prime
... no uniform way of predicting, for all primes p, which is the smallest primitive root modulo p. However, several results were known about the size of gp . In 1944, Pillai proved that there exist infinitely many primes p, such that gp > C log log p (where C is a positive constant). In particular, lim s ...
... no uniform way of predicting, for all primes p, which is the smallest primitive root modulo p. However, several results were known about the size of gp . In 1944, Pillai proved that there exist infinitely many primes p, such that gp > C log log p (where C is a positive constant). In particular, lim s ...
Flat primes and thin primes
... Some interesting subclasses of primes have been identified and actively considered. These include Mersenne primes, Sophie Germain primes, Fermat primes, Cullen’s primes, Wieferich primes, primes of the form n2 + 1, of the form n! ± 1, etc. See for example [14, Chapter 5] and the references in that t ...
... Some interesting subclasses of primes have been identified and actively considered. These include Mersenne primes, Sophie Germain primes, Fermat primes, Cullen’s primes, Wieferich primes, primes of the form n2 + 1, of the form n! ± 1, etc. See for example [14, Chapter 5] and the references in that t ...
Mathematical Reasoning: Writing and Proof
... course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are to help students: Develop logical thinking skills and to develop the ability to think mo ...
... course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are to help students: Develop logical thinking skills and to develop the ability to think mo ...
divisibility work sheet
... validate. We will finish off with an examination of how to test for divisibility by ...
... validate. We will finish off with an examination of how to test for divisibility by ...
Review Article On Bondage Numbers of Graphs: A Survey with
... bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination number greater than the domination number of G. The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considera ...
... bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination number greater than the domination number of G. The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considera ...
Full text
... As shown in [2], an escalator sequence is uniquely determined by its base, a1 = A1 , and any rational number other than 1 is the base of an (infinite) escalator sequence. In this paper, we consider the possibility of reversing the process. What can we deduce about an escalator sequence, given a sing ...
... As shown in [2], an escalator sequence is uniquely determined by its base, a1 = A1 , and any rational number other than 1 is the base of an (infinite) escalator sequence. In this paper, we consider the possibility of reversing the process. What can we deduce about an escalator sequence, given a sing ...
Review Sheet for Math 471 Midterm Fall 2014, Siman Wong Disclaimer: Note:
... In the special case of prime modulus we have the additional equivalence p - a ⇐⇒ gcd( p, a) = 1. This is totally false if p is NOT a prime! This is also a good place in the review notes to point out that the Fundamental Theorem of Arithmetic allows us to give yield another reformulation of GCD (even ...
... In the special case of prime modulus we have the additional equivalence p - a ⇐⇒ gcd( p, a) = 1. This is totally false if p is NOT a prime! This is also a good place in the review notes to point out that the Fundamental Theorem of Arithmetic allows us to give yield another reformulation of GCD (even ...
Basic Number Theory
... • Let a,b,c,n be integers with n≠0 (1) a≡0 (mod n) iff n|a (2) a≡a (mod n) (3) a≡b (mod n) iff b≡a (mod n) (4) a≡b and b≡c (mod n) → a≡c (mod n) (5) a≡b and c≡d (mod n) → a+c≡b+d, a−c≡b−d, ac≡bd (mod n) (6) ab≡ac (mod n) with n≠0, and gcd(a,n)=1, then b≡c (mod n) ...
... • Let a,b,c,n be integers with n≠0 (1) a≡0 (mod n) iff n|a (2) a≡a (mod n) (3) a≡b (mod n) iff b≡a (mod n) (4) a≡b and b≡c (mod n) → a≡c (mod n) (5) a≡b and c≡d (mod n) → a+c≡b+d, a−c≡b−d, ac≡bd (mod n) (6) ab≡ac (mod n) with n≠0, and gcd(a,n)=1, then b≡c (mod n) ...
Solutions - CMU Math
... After trying values, you will find that s = 2, 4, 16, or 18 are the only values possible. Thus, that are 4 ∗ 5 = 20 possible values of x ≤ 100. 14. (2004 AIME 2 10) Let S be the set of integers between 1 and 240 that contain two 1’s when written in base 2. What is the probability that a random integ ...
... After trying values, you will find that s = 2, 4, 16, or 18 are the only values possible. Thus, that are 4 ∗ 5 = 20 possible values of x ≤ 100. 14. (2004 AIME 2 10) Let S be the set of integers between 1 and 240 that contain two 1’s when written in base 2. What is the probability that a random integ ...
Cryptography and Number Theory
... Alice. It is not at all clear how to design such a function. In fact, when the idea for public key cryptography was proposed (by Diffie and Hellman2 ), no one knew of any such functions. The first complete public-key cryptosystem is the now-famous RSA cryptosystem, widely used in many contexts. To u ...
... Alice. It is not at all clear how to design such a function. In fact, when the idea for public key cryptography was proposed (by Diffie and Hellman2 ), no one knew of any such functions. The first complete public-key cryptosystem is the now-famous RSA cryptosystem, widely used in many contexts. To u ...
Cryptography and Network Security Chapter 4
... • if multiplication operation is commutative, we have a commutative ring • if multiplication operation has an identity and no zero divisors, it forms an integral domain ...
... • if multiplication operation is commutative, we have a commutative ring • if multiplication operation has an identity and no zero divisors, it forms an integral domain ...