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.

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.)

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.

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

This is surprising: A laminar flow induced in a viscous fluid confined in the gap between two rotating cylinders can be (to a large extent) reversible. The dyes here appear to mix, but in fact they’re being stretched into distinct spirals that can then be “unmade” by reversing the direction of the rotation.