Recent content by Vespero

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    Determinant of a Block Lower Triangular Matrix

    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...
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    Searching inside .zip files on internet

    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...
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    BFS Shortest Distance of Graph Calculation in Python

    Never mind, I got it. I needed to declare traversed[v] = True so that it wouldn't examine the starting vertex. Problem solved.
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    BFS Shortest Distance of Graph Calculation in Python

    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...
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    Prove infinitude of primes of form 4k+1 using properties of Legendre symbol (-1/p)

    Homework Statement Show that there are infinitely many primes 4k+1 using the properties of \left(\frac{-1}{p}\right). Homework Equations \left(\frac{-1}{p}\right) = \begin{cases} 1, & \text{if }p\equiv 1\ (mod\ 4), \\ -1, & \text{if }p\equiv 3\ (mod\ 4). \end{cases}...
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    Proving the Formula: $\sum_{a=1}^{p-2} \left(\frac{a(a+1)}{p}\right) =\ -1$

    Homework Statement Prove the formula \sum_{a=1}^{p-2} \left(\frac{a(a+1)}{p}\right) =\ -1 Homework Equations \left(\frac{xy}{p}\right) = \left(\frac{x}{p}\right) \left(\frac{y}{p}\right) \left(\frac{x}{p}\right) \equiv x^{\frac{p-1}{2}}\ (mod\ p) (Euler's Criterion) and other basic...
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    Calculation of Large Exponents Modulo a Prime

    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...
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    How many consecutive 1's are necessary for divisibility by a number?

    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...
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    Show -1 and +1 are only solutions to x^2 = 1 (mod p^2) for odd prime p.

    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...
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    Show -1 and +1 are only solutions to x^2 = 1 (mod p^2) for odd prime p.

    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...
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    Number Theory: Define G(n) and show property for any prime p

    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...
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    Number Theory: Define G(n) and show property for any prime 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...
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    Number Theory: Define G(n) and show property for any prime p

    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...
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    Number Theory: Define G(n) and show property for any prime p

    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...
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    Summation of Products of Binomial Coefficients

    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...
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