Activities
... share their solutions, using a variety of natural numbers greater than 1 but less than 100. Review the meaning of natural numbers and prime numbers. Through discussion, have students generalize that there is one and only one prime factorization for any natural number greater than 1 even though there ...
... share their solutions, using a variety of natural numbers greater than 1 but less than 100. Review the meaning of natural numbers and prime numbers. Through discussion, have students generalize that there is one and only one prime factorization for any natural number greater than 1 even though there ...
Pollard's p - 1 Method
... 1) Choose an integer k which is a multiple of most (or of all) of the numbers b ≤ B; e.g. k = B!. 2) Choose a (random) number a with 2 ≤ a ≤ n−2. 3) Compute r = rem(ak , n) by the power-mod method. 4) Compute d = (r − 1, n) by the Euclidean algorithm. 5) If d = 1 or d = n, start over (new a or new k ...
... 1) Choose an integer k which is a multiple of most (or of all) of the numbers b ≤ B; e.g. k = B!. 2) Choose a (random) number a with 2 ≤ a ≤ n−2. 3) Compute r = rem(ak , n) by the power-mod method. 4) Compute d = (r − 1, n) by the Euclidean algorithm. 5) If d = 1 or d = n, start over (new a or new k ...
Leap Frog Solutions 2015
... Problem 4. An arithmetic sequence of positive integers contains the numbers 6 and 122, though 122 is not the term directly after 6. There are no perfect squares in the sequence between the terms 6 and 122. What is the next number in the sequence after 122? Solution. The common di↵erence of the arith ...
... Problem 4. An arithmetic sequence of positive integers contains the numbers 6 and 122, though 122 is not the term directly after 6. There are no perfect squares in the sequence between the terms 6 and 122. What is the next number in the sequence after 122? Solution. The common di↵erence of the arith ...
Handout
... have the property that the distance between any two of them is an integer? Exercise: Find a set that contains 8 points that are integer distances apart. Question: Is it possible to find for any integer n ≥ 3, a set of n points in the plane such that no three are collinear and the distance between an ...
... have the property that the distance between any two of them is an integer? Exercise: Find a set that contains 8 points that are integer distances apart. Question: Is it possible to find for any integer n ≥ 3, a set of n points in the plane such that no three are collinear and the distance between an ...
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION
... If x = p/q ∈ Q, let us take a sequence {xn } of irrational numbers such that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking th ...
... If x = p/q ∈ Q, let us take a sequence {xn } of irrational numbers such that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking th ...
Ch 8 - ClausenTech
... The Fundamental Theorem of Algebra (Carl Gauss) For every polynomial of degree n > 1(with complex coefficients) there exists at least one linear factor. Another Theorem by Carl Friedrich Gauss Every polynomial of degree n > 1, (with complex coefficients) can be factored into exactly n linear factors ...
... The Fundamental Theorem of Algebra (Carl Gauss) For every polynomial of degree n > 1(with complex coefficients) there exists at least one linear factor. Another Theorem by Carl Friedrich Gauss Every polynomial of degree n > 1, (with complex coefficients) can be factored into exactly n linear factors ...
Ch8 - ClausenTech
... The Fundamental Theorem of Algebra (Carl Gauss) For every polynomial of degree n > 1(with complex coefficients) there exists at least one linear factor. Another Theorem by Carl Friedrich Gauss Every polynomial of degree n > 1, (with complex coefficients) can be factored into exactly n linear factors ...
... The Fundamental Theorem of Algebra (Carl Gauss) For every polynomial of degree n > 1(with complex coefficients) there exists at least one linear factor. Another Theorem by Carl Friedrich Gauss Every polynomial of degree n > 1, (with complex coefficients) can be factored into exactly n linear factors ...
