Applications of the Complex Roots of Unity - Rose
... right-hand side of expression (5) is m = pi. In this case, pin+1 is a factor of the right-hand side: therefore pinpi* is by definition the period of pin+1. One difference between factoring the Mersenne numbers {1, 5, 7, 15,…} and the sequence of integers {1, 2, 3, 4,…} is the first appearance of a p ...
... right-hand side of expression (5) is m = pi. In this case, pin+1 is a factor of the right-hand side: therefore pinpi* is by definition the period of pin+1. One difference between factoring the Mersenne numbers {1, 5, 7, 15,…} and the sequence of integers {1, 2, 3, 4,…} is the first appearance of a p ...
EQUIVALENCE RELATIONS Recall that "a equiv_n b" means n | b
... These are trivially obvious, so much so that is seems a waste of tine to even mention them. But they are important, and lead to this key notion: Definition: Given a set S, an "equivalence relation" on S is a set E of ordered pairs of elements x,y of
S having the "SRT" properties of
a. symmetry ...
... These are trivially obvious, so much so that is seems a waste of tine to even mention them. But they are important, and lead to this key notion: Definition: Given a set S, an "equivalence relation" on S is a set E of ordered pairs
Full text
... This description gives a very fast method for computing the coefficients a(m) recursively. Once we have computed them for 0 ≤ m < Fn we can immediately compute them for Fn ≤ m < Fn+1 using Proposition 1. Also, since the coefficient of xm in A(x) is equal to −1, 0 or 1 for all non-negative integers m ...
... This description gives a very fast method for computing the coefficients a(m) recursively. Once we have computed them for 0 ≤ m < Fn we can immediately compute them for Fn ≤ m < Fn+1 using Proposition 1. Also, since the coefficient of xm in A(x) is equal to −1, 0 or 1 for all non-negative integers m ...
On the Number of Prime Numbers less than a Given Quantity
... Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was rea ...
... Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was rea ...
2007 Mathematical Olympiad Summer Program Tests
... Prove that for all positive integers x and y, the numerator of fp (x) − fp (y), when written in lowest terms, is divisible by p3 . 3. Let n be an integer greater than 2, and P1 , P2 , · · · , Pn distinct points in the plane. Let S denote the union of the segments P1 P2 , P2 P3 , . . . , Pn−1 Pn . De ...
... Prove that for all positive integers x and y, the numerator of fp (x) − fp (y), when written in lowest terms, is divisible by p3 . 3. Let n be an integer greater than 2, and P1 , P2 , · · · , Pn distinct points in the plane. Let S denote the union of the segments P1 P2 , P2 P3 , . . . , Pn−1 Pn . De ...
Dynamical Sieve of Eratosthenes
... In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers N , where each number n represents a time t. In the same wa ...
... In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers N , where each number n represents a time t. In the same wa ...
The classification of 231-avoiding permutations by descents and
... These generating functions have recently turned up in a completely different context. In [8], Kitaev, Remmel, and Tiefenbruck studied what they called quadrant marked mesh patterns. That is, let σ = σ1 . . . σn be a permutation written in one-line notation. Then we will consider the graph of σ, G(σ) ...
... These generating functions have recently turned up in a completely different context. In [8], Kitaev, Remmel, and Tiefenbruck studied what they called quadrant marked mesh patterns. That is, let σ = σ1 . . . σn be a permutation written in one-line notation. Then we will consider the graph of σ, G(σ) ...
