これで２項定理Binomial Theoremが嫌いなKidsが好きになってくれたらね。
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Kidsが通う私塾の老先生は２項定理に興味をひきつけるためにネットから引き出してきたMichael Rose の論文。
私はHockyStickしか知らなかった。特にIntertwining petalsは６角形花で面白い！説明は老先生に任せよう。
Prime … num-ber things!
PhD Candidate, School of Mathematical and Physical Sciences,
University of Newcastle
theconversation.com/the-12-days-of-pascals-triangular-christmas-21479
If you add up every single number in the first n rows, you’ll get the nth Mersenne number (which is the number that falls 1 short of being 2 to the power of n).
Mersenne numbers are used at the cutting edge of mathematical research to find really large prime numbers, because they have a very interesting feature: if n is a prime number, then every now and then the nth Mersenne number will be prime as well.
For example, adding up all the numbers in the first 5 rows of Pascal’s triangle gives us the 5th Mersenne number, 31(which is 1 less than 2 to the power of 5).
Since 5 is a prime number, there’s a possibility that 31 might be a prime number too… and it just so happens that it is.
To date, the largest known Mersenne prime is (2 to the power of 20996011) minus 1 – a number with 6320430 digits!
老先生がKidsに計算させる。
第２段 2^2 - 1 = 1 + 1 + 1 = 3 Prime!
第３段 2^3 -1 = 1 + 1+ 1 + 1+ 1 + 2 = 7 Prime!
第７段 2^7 -1 = (2^5 -1) + 2(1 + 3 +10) + 2(1 + 6 +15) +20
= 31 + 32 + 44 + 20 = 127 !! Forth Mersenne Prime!
さすがのKidsも２項定理が面白くないと言わなくなったようである。ただ女の子が社会に出てこんな話がなんの意味があるのかと老先生を困らしたらしい。男女のBrainの違いがこんなところに現れる。
私は大学数学まで習ったがテイラー展開だδーε論法だと面白くない講義ばかりであった。歴史的な展開に欠けている講義はボイコットしてしまった記憶がある。
ｄｘや∬の記号の嫌いなKidsの話意によると老先生もJames Gregory、John Wallis, Newton などのイギリス数学やスイスのオイラーまでは青年の自分には価値あるがガウス（modなど余りにも恣意的、フランスのAdrien-Marie Legendreで充分）などのドイツ数学は陰気臭くて受け付けないらしい。
Leibnitzの記号的な寄与は認めるが、やはり微積の創始はNewtonであろうという。Wallisは奇術師のようなLeibnitzが如何にも自分が考えたような気分で論文を出版したので、「単なるNewtonからの剽窃ではないか」と諌めたというわけである。
もちろんMersenne 素数など聞いたこともなかった。その点、Kidsらは老先生の歴史的講義が聞けて幸せである。
npr.org/templates/story/story.php?storyId=102871918
Intertwining petals
Hockey-stick addition
Pick any counting number from along the first diagonal and square it. Then look at its two neighbours that lie deeper inside the triangle – they’ll always add up to that very same square number.
Powers of eleven
As an example, when squishing the 1-4-6-4-1 row together, we just end up with the number 14641. But when we squish the 1-5-10-10-5-1 row together, we split the 10s up into a 1 (which gets added to the number on the left) and a 0 (which stays put).
Upon splitting the first 10 this way, the row becomes 1-(5+1)-(0)-10-5-1. When the second 10 is split, the row ends up as 1-(5+1)-(0+1)-0-5-1, or 1-6-1-0-5-1, and we end up with the number 161051 (it’s a lot easier to do it than to describe it, trust me!).
Just as combining the numbers in a row by adding them gives us the powers of two, combining the numbers in a row by squishing them together like this gives us the powers of eleven! Also note that 11 to the power of 0 really wants to be equal to 1 as well…
On the twelfth day of Christmas, the triangle gave to me…