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20230816mathstuff

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20230816mathstuff [2023.08.16 20:12] – created Steve Isenberg20230816mathstuff [2023.08.16 20:17] (current) – Steve Isenberg
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 ====Math stuff for LCTG with Charlie==== ====Math stuff for LCTG with Charlie====
  
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 ===Approach=== ===Approach===
-Build up a-dics in small steps. +Build up p-adics in small steps. Maybe background steps:
-  * Background: +
   * Pythagorean Triples -- Charlie knows how to generate them   * Pythagorean Triples -- Charlie knows how to generate them
-  * number systems w/different bases+  * number systems w/different bases -- many know base 2, 8, 16; Muller uses base 3 (3-adics) and Rowland uses base 5 (5-adics)
   * harmonic series, summation, simple to prove; Muller uses the summation.  And he uses the series as a familiar illustration of convergence to the right to contrast the possibility of 10-adic convergence to the left.  And the same for 3-adic numbers.   * harmonic series, summation, simple to prove; Muller uses the summation.  And he uses the series as a familiar illustration of convergence to the right to contrast the possibility of 10-adic convergence to the left.  And the same for 3-adic numbers.
   * Diophantine equations   * Diophantine equations
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 +There is mention of modular arithmetic.  And mention that it is useful when testing Fermat's theorem for prime numbers (not his Last Theorem). Fermat's theorem says for any prime p if you take any relatively prime number n to the power p-1 and divide by p you will always get a remainder of 1.  Modulo arithmetic makes it possible to do in your head the calculation for three-digit prime numbers.  For example, I can do it for p = 101 and n = 2 .  Harder to do for n=3 or 4 or 5 ...).  
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 +Modular arithmetic is also useful for solving some Diophantine equations.  For example, find integers that solve 71x+37y = 3000. (I can do this without modular arithmetic.)
 +
 +And eventually we get to the point where we show how Muller solved x^2 + x^4 +x^8 = y^2 using 3-adics.
  
  
 [[https://www.youtube.com/watch?v=1UTjWy-vnOo|Mark Rober's Commencement Speech]] (20m) [[https://www.youtube.com/watch?v=1UTjWy-vnOo|Mark Rober's Commencement Speech]] (20m)
20230816mathstuff.1692241960.txt.gz · Last modified: by Steve Isenberg