Here’s the link

]]>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.”

]]>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.

]]>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).

]]>[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].

]]>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.

]]>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.

]]>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

]]>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.

]]>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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