Google Books Ngram

Pak’s history of the Catalan numbers mentioned an Ngram chart. But perhaps because I wasn’t reading carefully or possibly due to the fact that I was reading the print copy and did not access url pointing to the said chart, I did not find out about this tool until today.

Here’s the link

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Faraday

Compared to mathematics, physics is extremely well-funded. For example, I have attended many talks by Nobel Laureates compared to Fields Medalists. (Locally, I have attended talks by Serre, S.T. Yau and Pierre Louis Lions. I somehow missed talks by Smale and Ngo Bao Chao. Overseas, I have heard Bhargava and Tao lecture.) But back to physics, in addition to the Nobel Lauretes, there have been many public talks on quantum mechanics, cosmology and particle physics, in relation to the LHC.

The latest was on 25th August by David Gross. He mentioned that Faraday, as an answer to the question what was his discovery good for, was said to have answered “you will be able to tax it.” A quick search on the web would reveal that this story is probably apocrypha, but still a good story. It was only a day later via a tweeter feed (it’s GMT +8 here) that I noted Faraday passed away on 25th August 1867, 150 years ago to the day!

It was certainly not the first time I heard the Faraday joke and conceivably I might have heard it from the same speaker but it also triggered my memory of another similar story. Through the modern marvels of ipad-photography (I took a quick snap of the slide and the photo had the date), google calendar and the internet I tracked it down to a talk by John Ellis on 19th Jan 2012 on the LHC. (A few months before the Higgs boson was observed.) The conversion with M. Thatcher and Ellis as follows:

Thatcher: “What do you do?”
Ellis: “Think of things for the experiments to look for, and hope they find something different.”
Thatcher: “Wouldn’t it be better if they found what you predicted?”
Ellis: “Then we would not learn how to go further!”

This rings well with David Gross’ own take;”The most important product of knowledge is ignorance.”

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What are the chances that a bird will poop on my car?

This question was posed as a scenario of mathematics in everyday life. Some googling led to Randall “XKCD” Munroe’s post here. Borrowing similar ideas leads to my own computation below.
I park at an open air carpark for work. The easy assumptions are my car has a 4.3m x 1.75m footprint and I park for 8 hours a day. For data on birds, after some searching I found a 2016 article that estimates (provided I interpreted correctly) a total of 377 birds in 113 hectares of built-up area at another university in the country. So if we assume a similar bird population here and that birds poop uniformly everywhere once every hour,
[tex] \frac{377}{113 \times 10,000} \times 8 \times 4.3 \times 1.75 \approx 2\% [/tex]

In my opinion, the estimate is on the low side. In practical terms, it appears birds spend a large amount on time perched on trees. I also read that birds tend to clear their vowels to lighten their loads before they fly. So it is a catch-22 for me. I either park in the shade under trees and risk being bombarded or I park in the open and find myself a 40 degrees oven when I leave work on a typical sunny afternoon.

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Queen of Mathematics

Number theory has been called the Queen of Mathematics. Until some fifty years ago, it did not occur to anyone that number theory, especially the study of prime numbers, would have any immediate applications to business. More recently, the Queen has been relegated to be the object of a courtship, inspired by material gains, rather than awe. As a result, progress has been made in unexpected directions, which have required deeper investigations. — Papa Paulo

A.K.A. Paulo Ribenboim from his pseudo-novel-number theory text “Prime Numbers, Friedns Who Give Problems: A Trialogue with Papa Paulo” (p. 50).

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Triangular numbers modulo powers of 2 and its generalizations

Someone discussed with me an interesting problem that he was working on with his students. They found that the congruence
[tex] \frac{1}{2}X(X+1) \equiv a \pmod{n} [/tex]
has a solution for every [tex] 0 \le a < n [/tex] if and only if [tex] n =2^k[/tex].

My first instinct of course was to complete the squares for triangular numbers and reduce the problem to [tex] X^2 \equiv a \pmod{n} [/tex].
This turn out to work well for odd modulus and the solutions for triangular numbers and squares correspond. But when the modulus was a power of 2, completing the square would not work. A simple search found a few websites where the phenomenon was recorded and it seems a (perhaps original?) source is Knuth’s the Art of Computer Programming Volume 3, Section 6.4, Exercise 20. Knuth was talking about hashing but essentially the exercise is the above problem. I prefer to rephrase it as for every positive integer k, these two sets are identical:
[tex] \{ \frac{x(x-1)}{2} \pmod{2^k} \} = \{0, 1, \ldots, 2^k-1 \} [/tex]

Knuth gave a slick proof which can be easily adapted and generalized to the following:
For a prime [tex] p, 1 \le m < p, k \ge 1,[/tex]
[tex] \{ px^2+mx \pmod{p^k} \} = \{0, 1, \ldots, p^k-1 \} [/tex]

Proof: Suppose
[tex] px^2+mx \equiv py^2+my \pmod{p^k} [/tex]
[tex] (x-y) ( p(x+y)+m) \equiv 0 \pmod{p^k}[/tex]
Since [tex] p \nmid p(x+y)+m [/tex] we can conclude that
[tex] x \equiv y \pmod{p^k}[/tex].

So each of [tex]x = 0, 1, \ldots p^k-1 [/tex] gives rise to a different value of [tex] px^2+mx \pmod{p^k}[/tex].

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If there is some good inflammable stuff it will catch fire

Do not try to satisfy your vanity by teaching them great many things. Awake their curiosity. It is enough to open the minds, do not overload them. Put there just a spark. If there is some good inflammable stuff it will catch fire.

The quote appears at the end of chapter 14 of Polya’s Mathematical Discovery. Polya attributes the quote to Anatole France from Le jardin d’Epicure. Perhaps he translated the French into English. He further adds: There is a great temptation to paraphrase this passage: “Do not try to satisfy your vanity by teaching high school kids a lot of … just because you wish to make people believer that you understand it yourself …” Yet les us resist temptation.

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Catalan numbers

I really enjoyed reading Federico Ardila’s article in the Mathematical Intelligencer. Apparently there was a vote of 3030 members at an assembly of CUP (Not Cambridge University Press but the Candidatura d’Unitat Popular). The vote had to do with forming an alliance with another party and ultimately related to the independence of Catalonia. The amazing thing that happened was that the vote came out 1515 Yes and 1515 No.

The probability that a YES-NO vote of 2m persons ends up in a tie is [tex] \binom{2m}{m}/2^{2m} [/tex], closely related to the Catalan number [tex] \binom{2m}{m}/(m+1) [/tex]. I love the dual connections and of course Ardila did not fail to mention Stanley’s Enumerative Combinatorics. What I did not know was that Stanley even included a joke. Made my day.

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Truncatable Primes

A colleague asked about sequences of primes a(n) such that a(n+1) is obtained by appending a single digit (in base 10) to the right of a(n).
For example: 3, 31, 311 …

Some thinking lead to the conjecture that such sequences are of finite length and that it is possible to use an exhaustive search to find all of them. A natural question would be what is the longest possible sequence but I was unable to find any conclusive answer on the web. So I decided to write a simple (and not very efficient) recursion in maple to search for all such primes. Here’s my ugly code:

cat3prime:= proc(n)
local d, s, i; s:=n; d:=irem(n,10);
if isprime(s) then print(s); return(cat3prime(10*s+1));
else for i from d to 7 by 2 do
if isprime(s-d+i+2) then s:=s-d+i+2; print(s); return cat3prime(10*s+1); fi; od;
while (irem(s,10)=9) do s:=(s-9)/10; od;
if s=0 then return print(“search complete”);
else return cat3prime(s+2); fi; fi;
end proc;

The search yielded five sequences of length 8 and no other longer sequences:
2,23,233,2339,23399,233993,2339933,23399339
2,29,293,2939,29399,293999,2939999,29399999
3,37,373,3733,37337,373379,3733799,37337999
5,59,593,5939,59393,593933,5939333,59393339
7,73,739,7393,73939,739391,7393913,73939133

Only one of the above sequences appears as is on OEIS but with a more careful search, I found a sequence called right-truncatable primes.

https://oeis.org/A024770

Which contains 83 primes, where successively truncating one digit from the right still results in a prime. I verified that my own maple search also yielded 83 primes and I guess the two lists must be identical. See the wikipedia entry on truncatable primes https://en.wikipedia.org/wiki/Truncatable_prime

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Twin corrections

Today is the 20th anniversary of the passing of Erdős and I would like to make two corrections. I had always thought the accent on Erdős’ name was ö , html code &#246 but it is actually Hungarian, html code &#337. The second is the coffee quote which I had attributed to him. I realised my mistake a number of years ago but never got a chance to correct it online. Both errors were perpetuated in this post from 2004. Here is a quote from Erdős’ paper “Child Prodigies”

In Hungary, many mathematicians drink strong coffee, in fact Rényi once said “a mathematician is a machine which turns coffee into theorems.”

Correction done but sadly I am still not quite sure how to pronounce his name correctly.

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Bollobas on solving problems

What you should be terrified of is a blank sheet in front of you after having thought about a problem for a little while. If after a session your wastepaper basket is full of notes of failed attempts, you may still be doing very well. Avoid pedestrian approaches, but always be happy to put in work. In particular, doing the simplest cases of a problem is unlikely to be a waste of time and may well turn out to be very useful.

Bela Bollobas, from Advice to Young Mathematicians

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Logical order

The most efficient logical order for a subject is usually different from the best psychological order in which to learn it

William Thurston from his book Three dimensional geometry and topology.

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One-seventh ellipse

Fun fact. It is well known that
[tex] \frac{1}{7} = 0.\overline{142857} [/tex].

It turns out that if the repeating digits are taken in sequence as (x,y) pairs in the following manner to form six points:
(1,4), (4,2), (2,8), (8,5), (5,7), (7,1),
then these six points actually lie on an ellipse defined by the following equation.
[tex] 19x^2+36xy+41y^2-333x-531y+1638=0 [/tex]

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Lakatos on Discovery

Discovery does not go up or down, but follows a zig-zag path: prodded by counterexamples, it moves from the naïve conjecture to the premises and the turns back again to delete the naïve conjecture and replace it by the theorem. Naïve conjecture and counterexamples do not appear in the fully fledged deductive structure: the zig-zag of discovery cannot be discerned in the end-product.

From his Proofs and Refutations.

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Notes from ICME13

Gila Hanna mentioned the carpet proof of the irrationality of [tex]\sqrt{2}[/tex]. A little digging reveals that it was due to Tennenbaum (1950s) and popularized by Conway (1990s). The original proof appeared in a book but the simple idea is described in this paper “Picturing Irrationality” by Steven Miller and David Montague in the Mathematics Magazine (2012). What is more interesting is the expanded version of their paper on arXiv called “Irrationality from the Book”. One can only infer the amount of editing and changes that took place from submission to publication.

This is a quote from Guershon Harel that I am trying to piece together from memory.

Some of us are constructing open neighbourhoods* but these neighbourhoods are in no way compact.

By neighbourhoods, he was referring to the impact of mathematics education on the curriculum.

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Logic joke

Three logicians walk into a bar. The bartender says:”Do you all want a drink?”
First logician says “I don’t know.”
Second says “I don’t know.”
Third says “Yes!”.

Meta: Why is this a joke?

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Hypergeometric functions

Hypergeometric functions are one of the paradises of nineteenth century mathematics that remain unknown to mathematicians of our day. Hypergeometric functions of several variables are an even better paradise: they will soon crop up in just about everything — Gian Carlo Rota

From p. 231 of Indiscrete Thoughts.

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Freeman Dyson

on how mathematics is permanent while physics is ephemeral.

it’s the beauty of mathematics, as opposed to physics, that it’s forever. I published my selected papers recently in one volume, and I found out that when you publish your selected papers most of the physics is ephemeral, that you don’t want to publish stuff that was written 10 or 20 years earlier, but the mathematics is permanent. So essentially everything I’ve ever published in mathematics is there, whereas only about a quarter of what I published on physics was worth preserving.

The quote is taken from a interview from 1998 via Web of Stories. Watch the video directly here. By the way, thumbs-up for providing transcripts. Great site!

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Halmos on how to study

It’s been said before and often, but it cannot be overemphasized : study actively. Don’t just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Paul Halmos – I want to be a Mathematician: An Automathography (1985), p.69.

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Discovery

What you have been obliged to discover by yourself leaves a path in your
mind which you can use again when the need arises. — G. C. Lichtenberg

I am not familiar with Lichtenberg but he was apparently a German physictists from the 1742 to 1799.

Added quote from Project Euler, the context is sharing one’s solution.

Real learning is an active process and seeing how it is done is a long way from experiencing that epiphany of discovery. Please do not deny others what you have so richly valued yourself.

Added quote attributed to Plato, not quite about the magic of discovery but somewhat related.

Do not train a child to learn by force or harshness; but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each.

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COMC problem of the week

A nice problem from the 2015 archive filed under week 9, dated 27th October 2015.

Define the function [tex]t(n)[/tex] on the nonnegative integers by [tex]t(0)=t(1)=0, t(2)=1,[/tex] and for [tex]n>2[/tex] let [tex]t(n)[/tex] be the smallest positive integer which does not divide [tex]n[/tex]. Let [tex]T(n)=t(t(t(n))).[/tex] Find the value of [tex]S[/tex] if [tex]S=T(1)+T(2)+T(3)+⋯+T(2006)[/tex].

My answer is 1171. Here’s my solution.
It is easy to see that [tex] t(n)=2 [/tex] for odd [tex] n \ge 3 [/tex]. Correspondingly, [tex] T(n)=0 [/tex]. Also [tex]T(2)=0[/tex] so we just need to sum over all even numbers from 4 to 2006.
Claim: [tex] t(n) =p^k [/tex], for some prime.
Pf of claim: Suppose [tex] t(n)= ab , \gcd(a,b)=1, a, b > 1[/tex]. Since [tex] a < ab \Rightarrow a \mid n [/tex]. Likewise we have [tex] b\mid n \Rightarrow ab \mid n [/tex] which is a contradiction.
Now if [tex] t(n) [/tex] is odd, [tex] t(t(n))=2 \Rightarrow T(n)=1 [/tex]. On the other hand if [tex] t(n) = 2^k, k > 1[/tex], [tex] t(t(n))=3 \Rightarrow T(n)=2 [/tex].

So each term in the sum contributes either 1 or 2. Now [tex] t(n)= 16 [/tex] is impossible as we would have [tex] 5, 7, 11, 13 \mid n \Rightarrow n \ge 5005 [/tex].
In the case where [tex] t(n)= 8 [/tex] we have [tex] 4, 3, 5, 7 \mid n \Rightarrow n = 420(2k+1) [/tex]. Within the range of 4 to 2006, the only possible candidates are 420 and 1260.
Next for [tex] t(n)= 4 [/tex] we have [tex] 2, 3 \mid n \Rightarrow n = 6(2k+1) [/tex] and there are 167 choices in the range.
Finally, there are 1002 even numbers with a positive contribution to the sum, so
[tex] S= 1002 + 167 + 2 = 1171 [/tex].

A verification using mathematica

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