## Down the rabbit hole

The following has been a rather interesting journey – from a test question which seemed fine, to a subtlety which seemed easy, to a discussion with a number of different mathematicians about the nature of distributions, measure theory and regularisation. I will try and make it as clear as possible in the post below. Note, as mentioned in the comments, we have actually only found the solution to this problem for a constrained range of x, and not x $\in \mathbb{R}$. I didn’t want to complicate things any more than necessary here for first year students, but the comments are very important too.

In a recent class test, there was a question, written by me, which was not quite the question that I wanted to ask. It turns out that it does have an answer, but it’s not an answer that can yet be found by the means at the class’s disposal.…

## 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.…

## Nowhere Differentiable Functions

By: Jan Wuzyk

In this article I am going to show that nowhere differentiable functions do in fact exists and give a few examples, some of which are relatively modern. But first I’m going to try to answer a question that is, in my opinion, too rarely discussed in mathematics classes, ”Why do we care?”.

# Why we care

To answer this question we have to look into the history of mathematics. In 1821 Augustin-Louis Cauchy published his seminal book, Cours d’Analyse, this is generally recognized as the first serious attempt to put calculus on a rigours footing[Com][1] , mainly through introducing rigorous definitions of limits,continuity and differentiability among others, and the definitions that go with them2. This was also time the integral was defined as an area instead of simply as the antiderivative.

It should be noted that Cauchy by no means closed the issue of rigour in analysis but he provided a starting point.…

## The Newton-Raphson Method

How would you go about finding the value of $\sqrt{3}$ if you didn’t have a square root button on your calculator? Well, the most obvious thing might be to try some values, based on your knowledge of the square root function. You are being asked to find that x for which:

$x=\sqrt{3}$

or, in other words, that x, which, when squared  gives 3. We have to be a little careful here because we know that there will actually be two numbers which satisfy this (one positive, one negative), and we are interested in the positive one only.

So, we try some values, but we don’t do it randomly, we can see that because $1^2=1$ and $2^2=4$ that whatever number squared gives 3 must be between 1 and 2. We can try something called a binary intersection. This just means taking the values which we know bound the right answer (ie. we know that 1<x<2), and trying the number in the middle.…