Wrong, But Useful: Episode 25

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Our second anniversary, and it's a bit of a love-in. Sorry to disappoint.

  • Colin and his partner have a new baby, Fred!
  • Happy anniversary to us!
  • Number of the podcast: $2+\sqrt{3}$, because it comes up in the Lucas-Lehmer test for Mersenne primes.
  • A brief digression on the test, Mersenne primes and his list. The anecdote Colin refers to is about Frank Nelson Cole.
  • @johanges (Johan in real life) points us at a video on 3D printing as an education tool. Colin mentions Desmos. Dave wonders what other teaching tools could be printed.
  • Colin goes over to the dark side, gives a talk citing Gale's Law:

    "A pie chart is never the correct chart to use."

    and a pie chart by @notonlyahatrack (Will).
    Colin fails to remember the title of Tufte's The Visual Display Of Quantitative Information, which, in fairness, is a tricky title. He also wrote a blog post about statistics.

  • One of the other maths listening pleasures has its own twitter account: @BBCMoreOrLess, to which loyal follower Colin sucks up. A link to the YouGov profiler in which Wrong, But Useful doesn't feature.
  • Dave's been reading The Number Devil. Digression on the Goldbach conjecture (any even integer can be written as the sum of two primes). Relatedly, any number over five is the sum of three primes. For any $n > 1$, there's a prime number between $n$ and $2n$. Colin's been reading The Secrets of Creation by Matthew Watkins, Matt Parker's Things To Make And Do In The Fourth Dimension, and a book of mathematical tables from Harwell.
  • Dave's been listening to Taking Maths Further, with @stecks and @peterrowlett. Dave foreshadows a Cool Project involving Peter and some celebrity guests.
  • @peterrowlett and @realityminus3 (Elizabeth) sent us some nice messages!
  • @realityminus3 also challenges us about the answer to the puzzle from Episode 23; Dave thinks Colin got it right, and accuses Colin of being soft.
  • Gold stars to @joshurtree (Josh) and @notonlyahatrack (Will). Answers from last time: e.g.,

  • This week's puzzle: consider $P_n=(2+\sqrt{3})^n$, where $n$ is a positive integer. Show that as $n$ increases, $P$ gets progressively closer to an integer value, or explain why. For instance, $(2+\sqrt{3})^{42} \approx 1,051,470,266,970,439,230,972,301.999 999 999 999 999 999 999 9$ (25 consecutive 9s!)

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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7 comments on “Wrong, But Useful: Episode 25

  • Joshua Zucker

    I like the puzzle! Once I understood why it worked, I could check my understanding by seeing that the same thing is true of e.g. (27 + sqrt(731)) to large powers (at first I thought it might be something special about the sqrt(3)/2 being a nice trig function of a fraction of a circle, and wondered if your exponent being a multiple of 6 might matter, but then I saw what was really going on).

    I don’t know if you want spoilers here so I’ll say just one word: conjugate.

    Well, OK, and one other way of looking at it: you can show that (2+sqrt(3))^big is a root of x^2 – huge integer * x + 1 or something along those lines, and the roots of that are basically huge integer and 1/(huge integer). I thought that was interesting (because in my example with 731 the polynomial is a lot messier to figure out).

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