number theory – Mathemafrica

The 2018 South African Mathematics Olympiad — Problem 5

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. I have been writing about some of the problems from the senior paper from 2018. A list of all of the problems can be found here.

Today we will look at the fifth problem from the 2018 South African Mathematics Olympiad:

Determine all sequences $a_1, a_2, a_3, \ldots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \ldots$ , and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$ .

Since the sequence $a_1, a_2, \ldots$ is strictly increasing, we know that $a_n \geq n - 1$ for all positive integers $n$ . (We could prove this rigorously by induction.) This means that $a_{n - 1} + n \leq (a_n - 1) + (a_n + 1) = 2a_n$ for all $n$ , and so we know that $a_{n - 1} + n$ is equal to either $a_n$ , or to $2a_n$ for all positive integers $n$ . Perhaps we should try to figure out exactly when it is equal to $a_n$ , and when it is equal to $2a_n$ . If we knew, for example, that we always have that $a_{n - 1} + n = a_n$ , then we have reduced the problem to solving this recurrence relation.…

By Dylan Nelson| July 21st, 2019|Competition, English|1 Comment

The 2018 South African Mathematics Olympiad — Problem 3

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the week leading up to the contest, I plan to take a look at some of the problems from the senior paper from 2018. A list of all the posts can be found here.

Today we will look at the third problem from the 2018 South African Mathematics Olympiad:

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$ , and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$ .

In many number theory problems, it helps to consider the prime factors of the numbers involved, and in this problem we are in fact forced to do so because the question itself is about the prime factors of a number. When dealing with factors of a number or an expression representing some number, it of course helps to consider whether we can factorise the given expression.…

By Dylan Nelson| July 17th, 2019|Competition|1 Comment

Prime Suspects – The anatomy of integers and permutations, by Andrew Granville and Jennifer Granville, illustrated by Robert Lewis – a review

NB I was sent this book as a review copy.

From Princeton University Press.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…

By Jonathan Shock| July 9th, 2019|Book reviews, Reviews|1 Comment

The confusion about discontinuity

Whilst reading Mícheál Ó Searcóid’s book Metric Spaces, I found out about a nuance in the definition of continuity that I was not previously aware of, and something which may be taught incorrectly at high schools. M. Searcóid states that a function such as tan(x) is continuous (read the page here). The definition of continuity at a point is based on the fact that the function has a value at that point (if a function is continuous at x = a, then f(a) has a value in the expression |f(x)-f(a)|<ϵ). However, following M. Searcóid’s line of thought about continuous functions, it does not make sense to consider points at which a function is not defined. If we were asked to prove that the function is ‘discontinuous’ at a point, we would need to show that the condition for continuity at that point is false. And the negation* of the condition for continuity would not make sense at a point where the function value is not defined.…

By Aidan Horn| July 21st, 2016|Courses, Level: intermediate|2 Comments

On Convergent Sequences and Prime Numbers

Ever since Euclid first proved that there are infinitely many prime numbers, mathematicians have found ever more creative ways to prove the same result, and also various stronger theorems that imply it. Dirichlet’s Theorem, for example, states that if $m$ and $n$ are relatively prime integers, then there are infinitely many prime numbers of the form $mk + n$ for some integer $k$ . It is also known that the sum of the reciprocals of the prime numbers diverges, that the sum

$\displaystyle \sum_{\substack{p \leq n \\ p \text{ prime}}} \frac{1}{p} \sim \log(\log(n))$

and that the number of prime numbers less than $n$ is asymptotically equal to $\displaystyle \frac{n}{\log(n)}$ . In this blog post, we will continue this proud tradition by proving that there are infinitely many prime numbers which have your phone number somewhere in their digits, and which simultaneously have a prime number of digits.

To do so, we will look at the convergence of two different sums: that of the reciprocals of the primes with a prime number of digits, and that of the reciprocals of the natural numbers which do not contain your phone number amongst their digits.…

By Dylan Nelson| May 22nd, 2016|Level: intermediate, Uncategorized|2 Comments

A Mathematics Problem from the SBITC

The Standard Bank IT Challenge (SBITC) is an annual coding competition for undergraduate and honours students in South Africa. The contest consists of two rounds: a regional event named the “heats”, and the final. In the heats, teams of up to four students each compete against other teams from the same university, and the winning team from each of the nine top-performing universities is invited to the final round in Johannesburg. Each member of the winning team wins a prize, and the winning university receives a large cash prize, but students mostly participate for the enjoyment that is to be obtained in solving the problems, and to test their skills against a set of problems that is designed to challenge the participants.

This year, the final problem from the heats (which took place on Saturday, 16 May) was fairly mathematical in nature; or more-so than the other problems at least. Essentially, the problem asks the following:

We consider generalised Fibonacci sequences $T_n$ which satisfy the same recurrence relation $T_{n + 2} = T_{n + 1} + T_n$ as the Fibonacci numbers, but with the first two terms $T_1$ and $T_2$ being arbitrary positive integers.…