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Unrectangularizability / 2010-06-12
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There’s a way to describe prime numbers without mentioning divisibility (or multiplication, division, or any arithmetic at all actually) that I suspect might be easier to digest for people who aren’t the least bit thrilled by number theory.

We just need one preliminary concept. Consider an arrangement of objects into identical rows forming a rectangular grid, like this:

We’ll call it a rectangle as long as all of its sides are at least two objects long. This should all be pretty intuitive.

Note that in the previous example there are 35 objects in total. We could get this by multiplying, but if we’re trying not to use arithmetic, we could also get it by counting. So because we have a rectangle with 35 objects in it, we’ll say that 35 must be a rectangular number. All it means for a number to be rectangular is that if you have that particular number of objects, you can arrange them into a rectangle somehow. Because the rectangles must have sides at least 2, the smallest rectangular number must be 4:

Having now defined ‘rectangular number’ in terms that I optimistically suspect a preschooler could understand, we can get prime numbers with just this:

A prime number is any number greater than one that is not rectangular.

This is a negative definition for sure, but we can make it constructive by turning it into a method for showing a number is prime. This is as simple as checking all possible rectangle-shapes (or row-lengths). For example, to show that 37 is a prime, we look at what happens when we try to make a rectangle with every possible row length:

There’s always something leftover. So to be prime is to completely escape all attempts at rectangularization. I think it’s fascinating to consider very large primes, for which there would be millions, trillions, or however arbitrarily many plausible row sizes to make a rectangle from, and none of them work.

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