Generalizations of Carmichael numbers I
... lower bounds for λ(n) have been obtained in [28]. It is easily deduced from Korselt’s criterion that every Carmichael number is a product of at least three distinct primes (see e.g., [35]). It was unsolved problem for many years whether there are infinitely many Carmichael numbers. The question was ...
... lower bounds for λ(n) have been obtained in [28]. It is easily deduced from Korselt’s criterion that every Carmichael number is a product of at least three distinct primes (see e.g., [35]). It was unsolved problem for many years whether there are infinitely many Carmichael numbers. The question was ...
An amazing prime heuristic
... forms. But it would also imply there are infinitely many primes of the form 3 n − 1, even though all but one of these are composite. So we must be a more careful than just adding up the terms 1/ log n. We will illustrate how this might be done in the case of polynomials in the next section. As a fin ...
... forms. But it would also imply there are infinitely many primes of the form 3 n − 1, even though all but one of these are composite. So we must be a more careful than just adding up the terms 1/ log n. We will illustrate how this might be done in the case of polynomials in the next section. As a fin ...
Solutions to Assignment 7 22.3 Let S be the set of all infinite
... Solutions to Assignment 7 22.3 Let S be the set of all infinite sequences of 0s and 1s. Show that S is uncountable. Proof: We use Cantor’s diagonal argument. So we assume (toward a contradiction) that we have an enumeration of the elements of S, say as S = {s1 , s2 , s3 , . . .} where each sn is an ...
... Solutions to Assignment 7 22.3 Let S be the set of all infinite sequences of 0s and 1s. Show that S is uncountable. Proof: We use Cantor’s diagonal argument. So we assume (toward a contradiction) that we have an enumeration of the elements of S, say as S = {s1 , s2 , s3 , . . .} where each sn is an ...
DUBLIN CITY UNIVERSITY
... Describe how decryption is done in RSA. Describe a technique which can be used to implement this decryption more efficiently using the prime factors of the modulus, and use this technique to decrypt the ciphertext generated above. Solution: We want to calculate cd (mod pq) and can calculate this mor ...
... Describe how decryption is done in RSA. Describe a technique which can be used to implement this decryption more efficiently using the prime factors of the modulus, and use this technique to decrypt the ciphertext generated above. Solution: We want to calculate cd (mod pq) and can calculate this mor ...
Computability on the Real Numbers
... νQare any notation of Q equivalent to νQand any standard notation of the binary rational numbers Q2 := {z/2n | z ∈ Z, n ∈ N}. Computability concepts introduced via robust definitions are not sensitive to “inessential” modifications. It can be expected that they occur in many applications. On the oth ...
... νQare any notation of Q equivalent to νQand any standard notation of the binary rational numbers Q2 := {z/2n | z ∈ Z, n ∈ N}. Computability concepts introduced via robust definitions are not sensitive to “inessential” modifications. It can be expected that they occur in many applications. On the oth ...
Grade 7/8 Math Circles Modular Arithmetic 1 Introduction
... This is not an easy calculation, unless you have a calculator, but that defeats the purpose of modular arithmetic, which is to simplify complicated calculations. So, we propose an idea: What if we were to calculate each number with respect to that modulo before we add them together? Now this complic ...
... This is not an easy calculation, unless you have a calculator, but that defeats the purpose of modular arithmetic, which is to simplify complicated calculations. So, we propose an idea: What if we were to calculate each number with respect to that modulo before we add them together? Now this complic ...
Unit 2 Scholar Study Guide Heriott-Watt
... For two variables x and y, y is an explicit function of x if it is a clearly defined function of x This means that we can write y as an expression in which the only variable is x and we obtain only one value for y Thus y = 3x = 1 and y = sin (2x - 3) are examples of y as an explicit function of x. Q ...
... For two variables x and y, y is an explicit function of x if it is a clearly defined function of x This means that we can write y as an expression in which the only variable is x and we obtain only one value for y Thus y = 3x = 1 and y = sin (2x - 3) are examples of y as an explicit function of x. Q ...
On repdigits as product of consecutive Fibonacci
... where F0 = 0 and F1 = 1. These numbers are well-known for possessing amazing properties. In 1963, the Fibonacci Association was created to provide an opportunity to share ideas about these intriguing numbers and their applications. We remark that, in 2003, Bugeaud et al. [2] proved that the only per ...
... where F0 = 0 and F1 = 1. These numbers are well-known for possessing amazing properties. In 1963, the Fibonacci Association was created to provide an opportunity to share ideas about these intriguing numbers and their applications. We remark that, in 2003, Bugeaud et al. [2] proved that the only per ...
Document
... multiplicative inverse is unique • Suppose that the gcd of b and n is 1. (This assumption makes sense because of property 1). • Suppose that b has two multiplicative inverses mod n, say x1 and x2, such that bx1= 1 mod n and bx2 = 1 mod n. • Let q1 be the quotient when we divide bx1 by n and q2 be th ...
... multiplicative inverse is unique • Suppose that the gcd of b and n is 1. (This assumption makes sense because of property 1). • Suppose that b has two multiplicative inverses mod n, say x1 and x2, such that bx1= 1 mod n and bx2 = 1 mod n. • Let q1 be the quotient when we divide bx1 by n and q2 be th ...
Chapter 4 The Group Zoo
... “The universe is an enormous direct product of representations of symmetry groups.” (Hermann Weyl, mathematician) In the previous chapter, we introduced groups (together with subgroups, order of a group, order of an element, abelian and cyclic groups) and saw as examples the group of symmetries of t ...
... “The universe is an enormous direct product of representations of symmetry groups.” (Hermann Weyl, mathematician) In the previous chapter, we introduced groups (together with subgroups, order of a group, order of an element, abelian and cyclic groups) and saw as examples the group of symmetries of t ...
Number Theory for Mathematical Contests
... This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions s ...
... This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions s ...
Number Theory for Mathematical Contests
... We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Th ...
... We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Th ...
Topology of numbers
... forms in two variables with integer coefficients, where this last case uses John Conway’s marvelous idea of the topograph of such a form. A good part of the book is devoted to this last topic, and in fact an alternative title for the book might be “The Topography of Numbers". Prerequisites for reading ...
... forms in two variables with integer coefficients, where this last case uses John Conway’s marvelous idea of the topograph of such a form. A good part of the book is devoted to this last topic, and in fact an alternative title for the book might be “The Topography of Numbers". Prerequisites for reading ...
Lecture Notes on Discrete Mathematics
... 1. [Subset of a set] If C is a set such that each element of C is also an element of A, then C is said to be a subset of the set A, denoted C ⊆ A. 2. [Equality of sets] The sets A and B are said to be equal if A ⊆ B and B ⊆ A, denoted A = B. ...
... 1. [Subset of a set] If C is a set such that each element of C is also an element of A, then C is said to be a subset of the set A, denoted C ⊆ A. 2. [Equality of sets] The sets A and B are said to be equal if A ⊆ B and B ⊆ A, denoted A = B. ...
arXiv:math/0510054v2 [math.HO] 17 Aug 2006
... s(1 − ππ )(1 − 4ππ )(1 − 9ππ ) etc. before the pentagonal number theorem in this letter, and this is also discussed in the previous two letters in the Euler-Niklaus I Bernoulli correspondence. Euler also notes in this letter that the coefficients of the terms in the series 1 + 1n + 2n2 + 3n3 + 5n4 + ...
... s(1 − ππ )(1 − 4ππ )(1 − 9ππ ) etc. before the pentagonal number theorem in this letter, and this is also discussed in the previous two letters in the Euler-Niklaus I Bernoulli correspondence. Euler also notes in this letter that the coefficients of the terms in the series 1 + 1n + 2n2 + 3n3 + 5n4 + ...
New finding of number theory By Liu Ran Contents 1
... In ancient times, people can only calculate by hands. The natural number is from 1 to 10. Such as 11, 12, …, 21, 22, …,100, …, 1000, … is recorded as one number 10+. In 32 bit computer, the biggest number is 2^32 = 4294967296. Any number is more than 4294967296 recorded as 4294967296+. Similarly, In ...
... In ancient times, people can only calculate by hands. The natural number is from 1 to 10. Such as 11, 12, …, 21, 22, …,100, …, 1000, … is recorded as one number 10+. In 32 bit computer, the biggest number is 2^32 = 4294967296. Any number is more than 4294967296 recorded as 4294967296+. Similarly, In ...
Weighted Catalan Numbers and Their Divisibility Properties
... We are also interested in enumerating mathematical objects, such as flat paths. However, it may be hard to directly count these objects. Oftentimes, mathematicians take a set of objects that are harder to count and give a one-to-one correspondence to another set of objects which is easier to count. ...
... We are also interested in enumerating mathematical objects, such as flat paths. However, it may be hard to directly count these objects. Oftentimes, mathematicians take a set of objects that are harder to count and give a one-to-one correspondence to another set of objects which is easier to count. ...
21(4)
... problem may be related to the Fibonacci sequence; and, in fact, this is so. More generally, one may be interested in finding the probability distribution of the waiting time to find r heads in succession for the first time. As one may guess, these results contain generalized Fibonacci, Tribonacci, . ...
... problem may be related to the Fibonacci sequence; and, in fact, this is so. More generally, one may be interested in finding the probability distribution of the waiting time to find r heads in succession for the first time. As one may guess, these results contain generalized Fibonacci, Tribonacci, . ...