A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.

## Posts Tagged 'Number Theory'

### Numerical Coincidences

Published March 23, 2017 Occasional Leave a CommentTags: Number Theory, Recreational Maths

### Brun’s Constant and the Pentium Bug

Published March 9, 2017 Occasional Leave a CommentTags: Arithmetic, Euler, Number Theory

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:

Obviously, this could not happen if there were only finitely many primes.

### The Shaky Foundations of Mathematics

Published December 1, 2016 Irish Times Leave a CommentTags: Algorithms, Logic, Number Theory

The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].

The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.

Every number is interesting. Suppose there were uninteresting numbers. Then there would be a smallest one. But this is an interesting property, contradicting the supposition. By *reductio ad absurdum*, there are none!

This is the hundredth “That’s Maths” article to appear in *The Irish Times* [TM100, or search for “thatsmaths” at irishtimes.com]. To celebrate the event, we have composed an ode to the number 100.

The counting numbers that we learn as children are so familiar that using them is second nature. They bear the appropriate name **natural numbers**. From then on, names of numbers become less and less apposite.

### Random Harmonic Series

Published July 28, 2016 Occasional Leave a CommentTags: Analysis, Number Theory, Probability

We consider the convergence of the random harmonic series

where is chosen randomly with probability of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

### Lecture sans paroles: the factors of M67

Published June 30, 2016 Occasional Leave a CommentTags: Number Theory

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause.