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polymathprojects.org | ||
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xorshammer.com
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| | | | | Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy. This hierarchy defines sets of sentences $latex \Pi_i$ and $latex \Sigma_i$ for all nonnegative integers $latex i$. The definition is as follows: $latex \Pi_0$ and $latex \Sigma_0$ are both equal to the set of sentences $latex P$ such that... | |
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www.jeremykun.com
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| | | | | Problem: Prove there are infinitely many primes Solution: Denote by $ \pi(n)$ the number of primes less than or equal to $ n$. We will give a lower bound on $ \pi(n)$ which increases without bound as $ n \to \infty$. Note that every number $ n$ can be factored as the product of a square free number $ r$ (a number which no square divides) and a square $ s^2$. | |
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gilkalai.wordpress.com
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| | | | | Gowers, Green, Manners and Tao. They reminded me of the A-team of the 1980s television series: "If you have a problem, if no one else can help, and if you can find them, maybe you can hire... the A-Team." A conjecture of Marton, widely known as "the polynomial Freiman-Ruzsa conjecture" was certainly a holy grail | |
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mkatkov.wordpress.com
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| | | For probability space $latex (\Omega, \mathcal{F}, \mathbb{P})$ with $latex A \in \mathcal{F}$ the indicator random variable $latex {\bf 1}_A : \Omega \rightarrow \mathbb{R} = \left\{ \begin{array}{cc} 1, & \omega \in A \\ 0, & \omega \notin A \end{array} \right.$ Than expected value of the indicator variable is the probability of the event $latex \omega \in... | ||
