Category Archives: Number Theory

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 … Continue reading

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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 … Continue reading

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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 … Continue reading

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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 … Continue reading

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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 … Continue reading

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Prime (Car) Number

It has always been a regret of epsilon proportion that my car license plate number doesn’t have particular significance. For example it is is divisible by 7 and hence not a prime, unlike the fact that my apartment number and … Continue reading

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Fermat

was born on 17th August 1601. Incidentally [tex] 1601 = 1^2 + 40^2[/tex] is a prime.

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Translate

It’s slightly old but I only recently saw this article about how Google Translate make use of linear transformation. The new book on my desk eta products and theta series identities has the following quote in the preface. In der … Continue reading

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Nested radicals

I was trying to solve an olympiad type problem involving a nested radical of the form [tex] \sqrt{a+b\sqrt{r}}.[/tex] I had managed to discover that [tex] \sqrt{a^2- b^2r} [/tex] is an integer but it turned out the trick is to rewrite … Continue reading

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Math videos

“Speaker, I’d like to talk about twin prime numbers …” goes McNerney in the US Congress. This took place on 11 Feb 2014. More details may be found here. There is still hope in politics afterall. I would love to … Continue reading

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Prime Mystery

A nice puzzle from Aziz Inan on plus.maths.org The number N represents the first 6 digits of a special number. N consists of three prime numbers put side by side. These three prime numbers come xth, yth and zth on … Continue reading

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Quadratic reciprocity

– a proof due to Zolotarev and its connection to dealing cards. A list of 240 and counting published proofs of the theorema aureum.

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The beauty of number theory

The realization that she is posing a problem, or working on one, that Zeno or Euler could have posed might stimulate in a student an aesthetic motivation much more profound than the purported motivation factor associated with the “real-life” problems … Continue reading

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2013 Abel Prize

A well written note by Frenkel that attempts to explain Deligne’s work that won him this year’s Abel Prize.

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125th Anniversary of Ramanujan’s Birth

It was a privilege to be in Delhi, India to celebrate the 125th Anniversary of the birth of Ramanujan. It was a good conference with good talks by many speakers. Personally for me I didn’t get any new ideas or … Continue reading

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Is 0 even?

I first saw this from BBC news. The aftermath of hurricane Sandy resulted in a shortage of fuel and New York City had to implement an odd-even system. The following is taken from their press release: 1) Vehicles with license … Continue reading

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16 Plates

An interesting problem from the wordplay column of the New York Times. Can you arrange plates numbered 1 to 16, in a circle so the sum of adjacent pairs is always a perfect square? The quick answer is no, since … Continue reading

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Turing Doodle

I was supposed to wake up early before my two year old boy so that I could get some work done, but alas, I got distracted by the neat little Turing machine puzzle put up by the guys at google. … Continue reading

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It’s Pi Day today

A little creative tinkering yielded this mnemonic for [tex]\pi[/tex] Pi (apologies, latex died) to 15 places. “How I need a drink, alcoholic of course, after the heavy lectures involving Euclid’s algorithm” I’ve substituted “quantum mechanics” with “Euclid’s algorithm”, not exactly … Continue reading

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Hardy on Number Theory

The elementary theory of numbers should be one of the very best subjects for early mathematical instruction. It demands very little previous knowledge; its subject matter is tangible and familiar; the processes of reasoning which it employs are simple, general … Continue reading

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