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awwalker.com | ||
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algorithmsoup.wordpress.com
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| | | | | The ``probabilistic method'' is the art of applying probabilistic thinking to non-probabilistic problems. Applications of the probabilistic method often feel like magic. Here is my favorite example: Theorem (Erdös, 1965). Call a set $latex {X}&fg=000000$ sum-free if for all $latex {a, b \in X}&fg=000000$, we have $latex {a + b \not\in X}&fg=000000$. For any finite... | |
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mattbaker.blog
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| | | | | On Pi Day 2016, I wrote inthis post about the remarkable fact, discovered by Euler, thatthe probability that two randomly chosen integers have no prime factors in common is $latex \frac{6}{\pi^2}$. The proof makes use of the famous identity $latex \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$, often referred to as the "Basel problem", which is also due... | |
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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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xorshammer.com
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| | | Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive. The idea is that you construct a superset $latex \mathbb{R}^*$ which contains the reals and also some infinitesimals, prove that some statement holds of $latex \mathbb{R}^*$, and then use a general "transfer principle"... | ||
