Recent content by Vespero

Homework Statement Theorem. Let A be a k by k matrix, let D have size n by n and let C have size n by k. Then det \left(\begin{array}{cc}A&0\\C&D\end{array}\right) = (det A)\cdot (det D). Proof. First show that \left(\begin{array}{cc}A&0\\0&I_{n}\end{array}\right) \cdot...

Is anyone aware of a way to search through .zip files on the internet (such as in an archive site) without having to download and extract the files? For example, if I have a search phrase that may be in a .doc file inside a .zip file which is potentially stored with many other .zip files, I...

Never mind, I got it. I needed to declare traversed[v] = True so that it wouldn't examine the starting vertex. Problem solved.

Homework Statement I have this problem essentially figured out. There's just one tiny problem that I can't seem to solve. I'm supposed to write a function bfs(G, v) which takes a graph G stored as a dictionary, and a starting vertex v. The function bfs performs a breadth-first-search...

Homework Statement In my Number Theory class, we learned how to calculate the value of large exponents modulo primes using Euler's Theorem. I understand how to do this with exponents larger than the value of the totient function of the prime, which is p-1, but what about when the exponent is...

Homework Statement I'm trying to figure out how many successive 1's are necessary for a number composed solely of 1's to be divisible by another number x. For example, how many 1's are necessary for 1...1 to be divisible by 7? Simply performing the calculation shows that the first such...

So, x^2 \equiv 1\ mod\ p^n implies p^n|x^2 - 1. Factoring x^2 - 1, we have that p^n|(x+1)(x-1). Since (x + 1) - (x - 1) = 2, p^n|(x+1) and p^n|(x-1) only if p|2, which would imply that p=2. However, since we are dealing with odd primes, Euclid's Lemma requires that either p^n|(x+1) or...

Homework Statement Let p be an odd prime and n be an integer. Show that -1 and +1 and the only solutions to x^2 \equiv 1\ mod\ p^n. Hint: What does a \equiv b\ mod\ m mean, then think a bit.Homework Equations x^2 \equiv 1\ mod\ p^n \rightarrow x^2 = 1 + p^nk for k an integer. The...

Still looking for help on this one. Another possibility I've considered is the following: Let us assume that there are not infinitely many n>1 such that p does not divide G_n. Then there exists some integer n such that p divides G_n and p also divides G_k for all k>n. This implies that p...

I proved the equality of G_n with the product of the binomial coefficients, implying that G_n is an integer for every n>1, as the product of a finite number of integers is also an integer. Now I need to prove that for a given prime p, there are infinitely many n>1 such that p does not divide...

Okay, now. Computing the first few values yields the sequence 1, 2, 9, 96, 2500,... which are the products across consecutive horizontal rows of Pascal's triangle. 1 = 1 2 = 1*2*1 9 = 1*3*3*1 96 = 1*4*6*4*1 2500 = 1*5*10*10*5*1 and so on. In other words, it should be possible...

Homework Statement Define the numbers G_n = \prod_{k=1}^n (\prod_{j=1}^{k-1}\frac{k}{j}). (a) Show that G_n is an integer, n>1; (b) Show that for each prime p, there are infinitely many n>1 such that p does not divide G_n. Homework Equations The Attempt at a Solution I can see that the...

Homework Statement Find and prove a formula for sum{ (m1 choose r)(m2 choose s)(m3 choose t) } where the sum is over all nonnegative integers r, s, ant t with fixed sum r + s + t = n. Homework Equations The Attempt at a Solution I first attempted to find the number of combinations of r...

If A ∩ Y = Y, the complement of A in Y is the null set, which is simultaneously open and closed, I believe. This tripped me up, but I guess that all that matters is that the complement is open for the set to be closed, correct? If that's true, then A and Y could both be the same open...

Homework Statement Let Y be a subset of X. Give an example where a set A is closed in Y but not closed in X. Homework Equations A set is closed if its complement is open. A set is open if for every element x0 of the set, there exists an E > 0 such that U(x0;E) = {x|d(x,x0)< E} is...