Appreciation

How Many Primes Did We Miss?

The mathematics world is abuzz with the verification of a new largest known prime number. The number, $2^{57885161} - 1$ , is a Mersenne Prime, and has over 17 million digits. The previous largest known prime was $2^{43112609} - 1$ , which had a mere 12.9 million or so digits.

It’s interesting to note that, while it has been known for thousands of years that there are infinitely many primes, it is a challenge even today to find large ones.

It is also interesting to note how many primes were missed in jumping from the previous largest prime to this new largest prime.

A well-known and elegant result, Bertrand’s Postulate, states that there is always a prime number between n and 2n, for n > 1. For example, the prime 3 is between 2 and 4; the prime 5 is between 3 and 6; the prime 11 is between 10 and 20; and so on.

In particular, this says that there must be a prime between $2^{n}$ and $2^{n+1}$ , since $2 * 2^{n} = 2^{n+1}$ .

Thus, there must be a prime between $2^{43112609}$ and $2^{43112610}$ , and another between $2^{43112610}$ and $2^{43112611}$ , and so on!

Thus, there are at least 57,885,161 – 43,112,609 = 14,772,552 primes between $2^{57885161}$ and $2^{43112609}$ ! We can therefore safely say there are at least 14,772,551 primes between the largest and second-largest known primes!

Let’s hope it’s not another 4 years until we have a new largest prime on the block.

By MrHonner, 13 yearsFebruary 6, 2013 ago

Math Art: Hypotrochoids

This hypotrochoid art set was the perfect stocking stuffer for a math lover like myself! With a little practice you can quickly produce some really beautiful images.

By MrHonner, 13 yearsJanuary 27, 2013 ago

Appreciation Geometry Uncategorized

Fibonacci Flushers

While travelling in Europe, I became fascinated with the variety of toilet-flushing mechanisms I encountered. The typical toilet had a low-flow / high-flow option (which I imagine saves a great deal of water in the long run) , and a lot of creativity emerged in the way this two-flush system was implemented.

While documenting the many ways to flush, I found this rectangular model oddly familiar and appealing.

And then it hit me: this looks like the golden rectangle!

The golden ratio has long been used by artists and architects to create aesthetically pleasing work. It is, after all, the divine proportion. Could it be that these toilet-makers took their cues from the masters of art and math? I had to find out.

I dropped my image into Geogebra and took some measurements.

The total length of the rectangle divided by its height is around 1.71. So, it’s not quite the golden ratio, but it’s pretty close. This flush-design is about 90% divine, I suppose.

Maybe their next design will be closer to the perfect proportion.

By patrick honner, 13 yearsJanuary 18, 2013 ago

Appreciation Photography

Math Photo: Urban Buckyball

There’s something a bit spooky about this sculpture of nested, truncated icosahedra in Madson Square Park. The eerie lighting and coloring brings to mind an evil, robotic Jack-o-Lantern.

Appreciation Geometry

IKEA and the Packing Problem

You know who is really good at geometry? IKEA.

The efficient assemblage of triangles, rectangles, and irregular pentagons is impressive enough, but check out this clever use of an empty quarter circle!

An appropriately sized shim lets a couple of rectangles fit perfectly into that corner!

By MrHonner, 13 yearsJanuary 10, 2013 ago

How Many Primes Did We Miss?

Math Art: Hypotrochoids

Fibonacci Flushers

Math Photo: Urban Buckyball

IKEA and the Packing Problem

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