Vanishing Act

There is often peculiar humour about self-frustration. Consider, for example, a train of events which started outside the old Clarendon Laboratory, Oxford. I came across a dirty beaker full of water just when I happened to have a pistol in my hand. Almost without thinking I fired, and was surprised at the spectacular way in which the beaker disappeared. I had, of course, fired at beakers before; but they had merely broken, and not shattered into small fragments. Following Rutherford’s precept I repeated the experiment and obtained the same result: it was the presence of the water which caused the difference in behavior. Years later, after the War, I found myself having to lecture to a large elementary class at Aberdeen, teaching hydrostatics ab initio. Right at the beginning came the definitions — a gas having little resistance to change of volume but a liquid having great resistance. I thought that I would drive the definitions home by repeating for the class my experiments with the pistol, for one can look at them from the point of view of the beaker, thus suddenly challenged to accommodate not only the liquid that it held before the bullet entered it, but also the bullet. It cannot accommodate the extra volume with the speed demanded, and so it shatters.

— R.V. Jones, “Impotence and Achievement in Physics and Technology,” Nature 207:4993 (1965), 120-125

(When the Royal Engineers tried to use this trick to demolish a tall chimney, filling its base with 6 feet of water and firing an explosive charge into the water, “it succeeded so well that it failed completely”: The incompressible water flung the surrounding ring of bricks outward, leaving a foreshortened chimney suspended above in midair. This dropped down neatly onto the old foundation, upright and intact, “presenting the Sappers with an exquisite problem.”)

A Special Year

Reader Alan Jackson points out that 2021 is the product of consecutive primes: 2021 = 43 × 47.

It’s the first time this has happened since 1763 (= 41 × 43), and it won’t happen again until 2491 (= 47 × 53).

A quick way to find the factors is to note that 2021 is the difference of two squares, 452 – 22, so 2021 = (45 – 2) × (45 + 2).

(Thanks, Alan.)

Small World

https://commons.wikimedia.org/wiki/File:Statue_of_Little_Prince,_Kyiv.jpg
Image: Wikimedia Commons

In Antoine de Saint-Exupéry’s 1943 novella The Little Prince, the narrator encounters “a most extraordinary small person” whose planet is “scarcely any larger than a house.”

This led University of Ljubljana physicist Janez Strnad to consider the implications. If the radius of the prince’s planet were 64 meters and it had Earth’s density, then the weight of a prince with a mass of 30 kg would amount to 0.003 newtons, corresponding on Earth to the weight of a mass of 0.3 g. (If the planet had the density of an asteroid, his weight would be lower still.)

The planet cannot have an atmosphere, because the mean velocity of gas molecules is greater than the escape velocity.

If the prince moved faster than 80 millimeters per second he’d be sent into orbit around the planet; if faster than 11 centimeters per second he’d leave it altogether.

“He could overcome the limitations concerning his velocity by either binding himself with a rope to his planet or building a spherical shell around it,” Strnad concluded. “The human body adapts to weightlessness and astronauts have to perform special gymnastic exercises not to suffer on returning to the Earth. For the little prince, coming to Earth would be a serious adventure, were he not a fictitious character.”

(Janez Strnad, “The Planet of the Little Prince,” Physics Education 23:4 [1988], 224.)

Prime Circles

Is it always possible to arrange the numbers from 1 to 2n in a circle so that each adjacent pair sums to a prime?

  4  1
7      2
6      3
  5  8

As of 2016, solutions have been found for every case up to n = 106 — but no one has yet proven that it’s always possible.

Undefined

In 1941, biochemist Alexander Todd visited the Defence Research Establishment at Porton to see a demonstration of a new chemical weapon to be used against tanks. Afterward, “I proceeded to the bar where — believe it or not — there was a white-coated barman who was not only serving drinks but also cigarettes!”

I hastened forward and rather timidly said ‘Can I have some cigarettes?’

‘What’s your rank?’ was the slightly unexpected reply.

‘I am afraid I haven’t got one,’ I answered.

‘Nonsense — everyone who comes here has a rank.’

‘I’m sorry but I just don’t have one.’

‘Now that puts me in a spot,’ said the barman, ‘for orders about cigarettes in this camp are clear — twenty for officers and ten for other ranks. Tell me what exactly are you?’

Now I really wanted those cigarettes so I drew myself up and said ‘I am the Professor of Chemistry at Manchester University.’

The barman contemplated me for about thirty seconds and then said ‘I’ll give you five.’

Todd added, “Since that day I have had few illusions about the important of professors!”

(From his 1983 autobiography, A Time to Remember.)

Home Primes

Pick an integer greater than 1 (say, 14). List its prime factors in order from smallest to greatest (2 7), and then “paste” those factors together to create a new number (27). Apply the same procedure to that number, and keep going until you reach a prime number:

27 = 3 × 3 × 3 → 333
333 = 3 × 3 × 37 → 3337
3337 = 47 × 71 → 4771
4771 = 13 × 367 → 13367

13367 is prime, so that’s the end of that sequence: 13367 is the home prime of 14.

A home prime should exist for every positive integer, but finding them can be stupendously hard. The sequence starting with 49 has been pressed through 118 steps now without producing a prime; the search continues. Details are maintained at Patrick De Geest’s website World of Numbers.

De Bruijn’s Theorem

https://commons.wikimedia.org/wiki/File:De_Bruijn_theorem_coloring.svg

At age 7, F.W. de Bruijn found himself unable to pack a box measuring 6 × 6 × 6 quite completely with bricks measuring 1 × 2 × 4. The box had volume 216, so it might be expected to accommodate exactly 27 bricks, but he found there was no way to pack more than 26.

He mentioned this to his father, who happened to be mathematician Nicolaas Govert de Bruijn, and Nicolaas found that a “harmonic brick” (one in which the length of each side is a multiple of the next smaller side length) can be packed efficiently only into a box whose dimensions are multiples of the brick’s dimensions.

This can seen intuitively by imagining the 6 × 6 × 6 box filled with small colored cubes as shown here. No matter where it’s placed, each 1 × 2 × 4 brick must now displace an equal number of white and black cubes. But the box contains 112 white cubes and 104 black ones. So the task is impossible.

(Nicholas G. de Bruijn, “Filling Boxes With Bricks,” American Mathematical Monthly 76:1 [1969], 37-40.)

Fortunate Numbers

Multiply the first n prime numbers:

2 × 3 × 5 × 7 × 11 × 13 = 30030

Now find the smallest integer greater than 1 that will produce a prime number when it’s added to that product. In this example it’s 17:

30030 + 17 = 30047,

which is prime. This makes 17 a Fortunate number, named for Reo Fortune, the social anthropologist who first studied this. The first few Fortunate numbers are

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151 …

Are all Fortunate numbers prime? Fortune conjectured so, but whether it’s true remains an open problem.

The Clever Way

When I give talks on factoring, I often repeat an incident that happened to me long ago in high school. I was involved in a math contest, and one of the problems was to factor the number 8051. A time limit of five minutes was given. It is not that we were not allowed to use pocket calculators; they did not exist in 1960, around when this event occurred! Well, I was fairly good at arithmetic, and I was sure I could trial divide up to the square root of 8051 (about 90) in the time allowed. But on any test, especially a contest, many students try to get into the mind of the person who made it up. Surely they would not give a problem where the only reasonable approach was to try possible divisors frantically until one was found. There must be a clever alternate route to the answer. So I spent a couple of minutes looking for the clever way, but grew worried that I was wasting too much time. I then belatedly started trial division, but I had wasted too much time, and I missed the problem. …

The trick is to write 8051 as 8100 – 49, which is 902 – 72, so we may use algebra, namely, factoring a difference of squares, to factor 8051. It is 83 × 97.

— Carl Pomerance, “A Tale of Two Sieves,” Notices of the AMS 43:12 (December 1996), 1473-1485