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Category Archives: Uncategorized
The 15Puzzle and the symmetric group
Almost 150 years ago the game 15puzzle was invented and quickly gain lots of popularity. The rules are quite simple – you start with a board with 15 numbered pieces as in the image below, where the pieces are ordered … Continue reading
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Breaking up Pythagorean Triples
One of the thing I really love in mathematics is how different subjects, which at least when starting to learn mathematics seem far apart, can come together in interesting ways. While the first reaction of many people might be that … Continue reading
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Tagged gaussian integers, lattice, Pythagoras, shortest vector problem
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Mediants in Mathematics
In the previous post we learned about the mediant, which is the “wrong” way to add rationals : . While this operation is not well defined, and depends on the presentation of the rational numbers and not just their values, … Continue reading
On one weird rational average
One of the things that I love about Mathematics, is how one can come across a mathematical problem or a phenomenon which you can look at from several points of views, each one different, yet they all combine together to … Continue reading
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Counting primes in arithmetic progressions
We have now seen in the previous two posts a few results about prime numbers. In the first one, we saw why primes are so important and used Euclid’s proof to show that there are infinitely many primes. Moreover, we … Continue reading
Summing the prime reciprocals
In the previous post we saw why primes are so important, and used Euclid’s proof to show that there are infinitely many primes. We further conjectured that not only there are infinitely many primes, they are also “nicely” distributed among … Continue reading
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Improper integrals and periodic functions
The idea for this post came from a question I saw in a math help forum about improper integrals. While this problem has a very simple solution using basic tools in integral calculus, I want to show a more geometric … Continue reading
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Radars and the Chinese Remainder Theorem
The radar is a detection system that was developed before and during World War II for military uses, though by today it has many other applications including, for example, astronomical and geological research. The name radar is an acronym for … Continue reading
Billiard tables – and what is mathematical research
Mathematical research is something that most people don’t really understand. They can imagine someone in a lab mixing chemicals or doing experiments with some scientific machinery, but mathematical research? The goal of this post is to share with you a … Continue reading
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Lattice parametrization
We came to the point where we have already seen how lattices appear naturally in problems arising from number theory. In this post we construct a nice space which parametrize the set of all lattices of a certain dimension, with … Continue reading
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Tagged Homogeneous spaces, Hyperbolic space, Lattices, SL_2(Z)
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