Today, whilst at work, a colleague asked me what the largest prime number I know of is. I responded half-seriously with "seventeen" and then, like all of the best conversations had between humans, we steered into a good bout of prime number-related banter. One of the questions that came up was this: "Is there a prime … Continue reading Primes Aplenty

# Lessons on Lessening

During your school years, you probably would have seen this equation: $latex 3^2 + 4^2 = 5^2.$ Your teacher most likely would have shown you this whilst rambling on about triangles or something. And maybe, if you were well behaved and pretended to be interested, you might have seen this one: $latex 3^3 + 4^3 + … Continue reading Lessons on Lessening

# The Desk Calendar

I came across a problem some time ago in Martin Gardner's The Colossal Book of Short Puzzles and Problems. It's nice because, well, you'll see. Anyway, give it a go, and I'll post the solution in the next few days.

# Bucking the STEM trend

If you're doing a science degree (or maybe you're just bored on a Friday night), remove your nose from its standard in-textbook position and stick it instead into the Mapping Australian higher education 2016 report. The key points for you are probably these: "Only 51 per cent of the science graduates looking for full-time work had … Continue reading Bucking the STEM trend

# Triangles and Squares

Here's a quick one for you. Consider the sequence of triangular numbers $latex 1, 3, 6, 10, 15, 21,28,36,45, \ldots$ and the set of square numbers $latex 1, 4, 9, 16, 25, 36, 49,\ldots$ It seems that there are some numbers which are both squares and triangles! For example, 1 and 36 get to be both … Continue reading Triangles and Squares

# The Harmonic Hurdler

Here's a classic exercise in elementary number theory. Consider a (mathematical and thus constrained to a point) rabbit who starts at zero on the real number line and starts to jump towards her right. Her first jump is of length 1, her second jump is of length $latex 1/2$, her third jump is of length … Continue reading The Harmonic Hurdler

# Martin Gardner’s Beer Problem

The following problem comes out of Martin Gardner's book Wheels, Life and Other Mathematical Amusements: On a picnic not long ago Walter van B. Roberts of Princeton, N.J., was handed a freshly opened can of beer. "I started to put it down," he writes, "but the ground was not level and I thought it would be well to … Continue reading Martin Gardner’s Beer Problem

# e^pi > pi^e

Let's prove that $latex e^{\pi} > \pi^e$ without a calculator. If you haven't seen this before, give it a try before you read any further. Consider the function $latex f(x) = x/\ln(x)$ on the interval $latex (1,\infty)$. This function shoots off to positive infinity as $latex x$ tends towards either endpoint of the domain. Let's … Continue reading e^pi > pi^e

# Limits and Towers

Consider the problem of finding the following limit:$latex \lim_{x \rightarrow 0} x^x&s=2$It's actually not too bad. We can write$latex x^x = e^{ x \ln x}&s=2$and bring the limit into the exponent (as exponentiating is continuous) to get that$latex \lim_{x \rightarrow 0} x^x = e^{\lim_{x \rightarrow 0} x \ln x}&s=2$From here, all we need to do … Continue reading Limits and Towers

# Borwein Integrals

Feel up to seeing a bit of mathematical magic? See here.