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7stones.com | ||
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www.quantamagazine.org
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| | | | | A team of mathematicians has solved an important question about how solutions to polynomial equations relate to sophisticated geometric objects called Shimura | |
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www.jeremykun.com
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| | | | | Problem: Prove there are infinitely many prime numbers. Solution: First recall that an arithmetic progression with difference $ d$ is a sequence of integers $ a_n \subset \mathbb{Z}$ so that for every pair $ a_k, a_{k+1}$ the difference $ a_{k+1} - a_k = d$. We proceed be defining a topology on the set of integers by defining a basis $ B$ of unbounded (in both directions) arithmetic progressions. That is, an open set in this topology is an arbitrary union of arithmetic progressions from $ -\infty$ to $ \infty$. | |
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thatsmaths.com
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| | | | | There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and... | |
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njwildberger.com
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| | | We are supposed to have a very clear idea about the `laws of logic'. For example, if all men are mortal, and Socrates is a man, then Socrates is mortal. Are there in factsuch things as the "laws of logic"? While we can all agree that certain rules of inference, like the example above, are... | ||
