Professor Buchanan hurries up the corridor towards the assembled sixth-years. “I always forget that the stairs don’t just move but, sometimes, steps simply vanish! I was stuck between the third and fourth floor for ten minutes. Anyway, what are you all doing out here? In… In…It’s nice and warm inside the classroom.”

Introduction

As you may recall, last class we discussed a few properties of one-digit numbers. However, there are certain overarching classes of numbers that may also impact the traits associated with a given number.

Today, we will be discussing these larger groups - or, as formally stated in Arithmantical terms, supranumerical groups. There are many groups of interest, but, for the sake of simplicity, we will discuss the three most relevant and intuitive of these: primes, squares and triangular numbers.

Prime Numbers

By definition, a prime number is any natural number that can only be divided by one or by itself to generate a new natural number. For example, the number 11 is considered prime because it produces a natural number when divided by one (11/1 = 11) and by itself (11/11 = 1), but dividing it by any other number results in a non-natural rational number instead (11/2 = 5.5, for example).

It should be noted that these divisors MUST be distinct from one another, which is the reason why the number one is neither prime nor composite (i.e., the opposite of prime, a number that can be evenly divided by a number that is not one or itself).

Before we delve into the magical properties of primes, let us discuss a bit about a few properties that Muggles are equally aware of. First and foremost, there is one prime that can be considered very special - the number two, which is the very first prime, is special because it’s the only even prime number. By definition, an even number is any number that is divisible by two; this means that any even number larger than two is divisible by a factor other than one or itself, making it composite.

The number two is also very relevant in the composition of the so-called Mersenne primes. In simple terms, if you multiply the number two many times (in a process called exponentiation, which will be discussed in further detail in the next class) and subtract one from the result, there are good chances you’ll find a prime number. For example, (2 * 2) - 1 = 3 is prime, (2 * 2 * 2) - 1 = 7 is prime and (2 * 2 * 2 * 2 * 2) - 1 = 31 is prime. This is not a rule set in stone, though: (2 * 2 * 2 * 2) - 1 = 15 is not prime, as it can be divided by both 3 and 5.

Another interesting property of primes is that, for all prime numbers larger than three, they are exactly one number away from a multiple of six. For example, take the first ten primes after three - in other words, the set {5, 7, 11, 13, 17, 19, 23, 29, 31, 37}. If you take a closer look, all of these numbers are one step away from the first multiples of six, listed in the set {6, 12, 18, 24, 30, 36} for an easy comparison. Although there’s a mathematical reason for this phenomenon, we will not go over its proof for the time being.

Let’s talk about the magical properties of primes instead. First of all, primes are very destructive numbers, very suitable for combat applications and other sorts of damage. Many associate that with the fact that primes deliberately avoid the healing properties held by six and its multiples. Others associate that trait with the fact that primes generate rational, or broken, numbers when divided by anything other than themselves or one.

Secondly, primes are excellent in applications regarding secrecy and concealment, particularly when dealing with enchantments that protect information; perhaps the Fidelius Charm or the Obscuring Charm come to mind. In fact, it seems like Muggles appear to be aware of that property too, even if subconsciously, as the use of prime factors is used in Muggle cryptography (i.e., the science of studying techniques that protect communication so information cannot be accessed by a third-party). Arithmancers defend that this is the case because primes can only be properly “connected” to the numbers 1 and itself, representing both the speaker and the intended listener.

The last magical property of primes is its apt use for isolation purposes, such as spells that restrain a target or battle strategies devised to disrupt coordination between different groups. This, too, seems to be connected to the solitary nature of primes and their desired intent to not interact with the rest of the numbers.

Square Numbers

If prime numbers can be considered solitary and destructive, then square numbers could potentially be regarded as the absolute opposite of that. By definition, a square number is generated by multiplying a whole number by itself - which means that it can never be prime! If you think closely about it, a square number must be divisible by at least one, itself, and the number that was multiplied twice to generate the square number. As a concrete example, nine (or 3 * 3) cannot be prime because it’s divisible by one, three and nine.

Another very interesting property of square numbers is that they always retain the parity of the original number multiplied. For example, if you multiply five (an odd number) by itself, you will get 25 - a square that is also an odd number. The same holds true for four (an even number), which nets 16 (an even square) when multiplied by itself.

Speaking of four, if you pay close attention you’ll realize that square numbers are either a multiple of four (if the square is even) or exactly one number above a multiple of that digit (if the square is odd). This too can be explained through a formal proof, but that will not be discussed at this moment of your education.

The last mundane fact I’d like to mention about square numbers is that they are always positive, even when you multiply negative numbers! By definition, multiplying two negative numbers gives a positive number - more on that in our next class, but you can take me on my word for the time being - and, for that reason, square numbers are never going to be negative. It’s possible to multiply the same number twice to get a negative value if we use imaginary numbers, but this contradicts the definition of a square number, as we need to make sure to use whole numbers instead.

So, what are the magical properties of square numbers? Firstly, they are very social numbers and want to interact with those that are similar to them; it’s not surprising, therefore, that they are great candidates to be used in speeches devised to influence others and spells that deal with human emotions. Square numbers are also very artistic and helpful with creative endeavors - it’s not a coincidence, for example, that the famous singer Celestina Warbeck, for example, has 9 and 16 letters in her first and full name respectively - both numbers are perfect squares!

There’s a final property that you might find of particular interest, especially if you plan to open your own company in the future: since squares are always close to four - which, as you may recall, represents the material world and the physical aspects of the universe - and are generated by multiplying a number with itself, these numbers are excellent when used for inviting prosperity and abundance into your life. You should not be surprised by the fact that Gringotts has nine letters in its name, and so do many other financial institutions and large companies!

Triangular Numbers

At last, triangular numbers! Unlike the groups above, triangular numbers are compatible with both square numbers and prime numbers. Nevertheless, there is only one number that is both prime and triangular: three.

Before we continue that discussion, though, let’s start with the definition of a triangular number. There are two ways to define the concept: we can either use a visual argument or a mathematical one in order to explain what a triangular number is. From the visual argument, a triangular number is any natural number generated by piling up items, so that every item that is not on the bottom row is supported by two items below it. This is what gives triangular numbers their very characteristic shape and name!

Triangular numbers may also be defined mathematically. In simple terms, they are the sum of all consecutive¹ digits from 1 to any other natural number - for example, zero, one, three and six are the first four triangular numbers because they are equivalent, respectively, to 0, (0 + 1), (0 + 1 + 2) and (0 + 1 + 2 + 3).

Let us go back to our previous topic. There is only one triangular prime number - three - but many triangular square numbers! The first four examples of numbers that are both triangular and square are 0, 1, 36 and 1225. You may have noticed that these numbers increase in value very quickly; that is because numbers that are both triangular and square are very rare, which makes them all the more valuable in Arithmancy.

There is yet another important property when it comes to triangular numbers: by adding two consecutive triangular numbers, we are able to create a square number. For example, recall the set of the first four triangular numbers, {0, 1, 3, 6}. Adding one to three nets four, a square number, and so does adding three to six, which nets nine. If you’d like a more visual way of looking at it, see how the two triangles complete a full square when placed next one another? This is why consecutive triangular numbers always add up to a square!

There are two main magical uses for triangular numbers according to modern Arithmancy studies. The first of them, in fact, is strongly connected to its shape and the number three: triangular numbers are considered very stable and excellent when used to craft long-lasting magical artifacts, particularly if used to support the base spell in a spellcasting web. You will study the concept of a spellcasting web in your N.E.W.T. level Charms more in-depth - in the meantime, all you need to know is that the use of triangular numbers may increase the stability of your weave.

Another excellent use for triangular numbers can be seen in runic spells. You may have noticed that most of the simpler runic spells tend to use three glyphs, as they commonly generate the most long-lasting forms of magic. This is particularly true when we consider the fact that the number three is both prime and triangular, as well as the fact that many runic spells are designed as traps or attacks on those that try to invade a protected space.

Closing

Well, there you have it - this was our first foray into the most basic supranumerical Arithmantical groups. Although there are many other categories of numbers - such as palindromes (numbers that are read the same way front to back as well as back to front), happy numbers (numbers that add up to one when we add the square of its digits successively until we get a single-digit number) and Fibonacci numbers (numbers generated by adding the two previous digits in a sequence, with the sequence starting with the elements zero and one) - these numbers’ traits are out of the scope of this year. Keep up with your studies and you may come across them in your Arithmantical studies!

Dismissed.

¹ Consecutive Numbers: two integers that follow each other in a sequence. For example, 16 and 17 are consecutive natural numbers, as 17 immediately follows 16. 7 and 11 are consecutive prime numbers, as 11 immediately follows 7 in the prime number list.

Original lesson written by Professor Vaylen Draekon
Improvements made by Professor Epona Salvatrix and Professor Calum Buchanan
Image credits here, here, here and here