Chapter 1: The Hidden Treasure of Mathematics

A few nights ago, while relaxing in bed, I found myself engrossed in various lectures and articles. At that moment, I resolved to watch just one more math video before calling it a night. Little did I know that the video I chose would leave me wide awake for hours. The reason? I was astounded to learn about a remarkable mathematical theorem that I had never encountered before.

Despite holding a Master's degree in mathematics and dedicating my studies to numerous mathematical disciplines—about 40 in total—I had somehow missed this theorem entirely. This suggests that it isn't commonly taught. Even more bewildering is that this theorem remained unnoticed for over 2500 years, not due to its complexity but rather because the underlying pattern had yet to be discovered.

The Discovery

The mathematics behind this theorem is so straightforward that even schoolchildren could grasp it, yet there is still much we do not understand about it. The stunning fact remained concealed from antiquity until 1951, when mathematician Alfred Moessner made a groundbreaking revelation.

To illustrate this phenomenon, let’s consider the partial sums of odd natural numbers:

1 = 1,

1 + 3 = 4,

1 + 3 + 5 = 9,

1 + 3 + 5 + 7 = 16,

1 + 3 + 5 + 7 + 9 = 25.

Can you recognize the pattern? These sums correspond to square numbers: 1 = 1², 4 = 2², 9 = 3², and so forth. The ancient Greeks were aware of this relationship and even provided a brilliant visual proof.

Moessner's approach was to reinterpret this fact from a slightly different angle, framing it in a more constructive and almost algorithmic manner. His method is as follows:

Start with all natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…

where the ellipsis indicates that the pattern continues indefinitely. Next, remove every second number to yield the odd numbers:

1, 3, 5, 7, 9, …

Calculating the partial sums gives us the squares: 1, 4, 9,...

While this information is familiar, the generalization that follows is what makes it exciting. Now, consider all natural numbers again, but this time remove every third number, resulting in the sequence:

1, 2, 4, 5, 7, 8, 10, 11, 13, …

Calculating the partial sums leads to a new sequence:

1, 3, 7, 12, 19, 27, 37, 48, 61,…

Removing every second number from this sequence gives:

1, 7, 19, 37, 61,…

Taking partial sums once more results in:

1, 8, 27, 64, 125,…

Do these numbers ring a bell? They correspond to our cubes: 1 = 1³, 8 = 2³, 27 = 3³,…

This method can be visualized as follows:

This pattern continues. By removing every n-th number from the natural numbers, taking partial sums to form a new sequence, and then systematically removing numbers, we eventually arrive at the powers of n.

How had I never come across this before? It is so simple and beautiful.

Exploration of Further Patterns

But we are just scratching the surface! There is so much more to discover. For example, if we begin by eliminating other intriguing sequences, what new sequences might emerge? Is there a more general pattern to observe?

In the image above, removing triangular numbers from the natural numbers forms small triangles of numbers. By continuing to eliminate numbers in the same systematic fashion, we derive the sequence 1, 2, 6, 24, 120, … which corresponds to factorials: 2 = 2×1, 6 = 3×2×1, 24 = 4×3×2×1, etc.

Interestingly, triangular numbers can be expressed as the sum of the first n natural numbers, while factorials are the product of those numbers. This is no coincidence.

Moreover, if we remove numbers of the form n, 2n, 3n, 4n, … for n > 1, the output will yield 1^n, 2^n, 3^n, 4^n, … indicating an exponential relationship between the input and output. It appears that in general, if we multiply each number in the input sequence by a constant m, the output sequence elements correspondingly rise to the power of m.

Experimentation

To further explore these concepts, I developed a Python program capable of generating these sequences based on different input sequences of numbers to be removed. I even experimented with operations beyond addition.

For instance, if we start by excluding square numbers (i.e., numbers of the form n²), we end up with the sequence 2, 12, 144, 2880, … which corresponds to numbers of the form n! (n+1)!.

Conversely, if we remove the sequence of numbers 2, 6, 12, 20, 30, 42,… (which follows the formula n(n+1)), the resulting sequence is 1, 4, 36, 576, 14400,… which corresponds to (n!)².

This dual relationship is fascinating, particularly since numbers of the form n(n+1) are simply triangular numbers multiplied by 2.

Using the program, I uncovered several other intriguing relationships, such as:

• If the input sequence is {2^k} = {1, 2, 4, 8, 16, 32, …}, the output seems to be {3, 8, 60, 3456, 11612160,…}—the number of independent sets in the binomial tree of order n.
• Removing pentagonal numbers (numbers of the form n(3n-1)/2) yields a sequence of numbers of the form (n!)²(n+1)!.
• If the input sequence comprises tetrahedral numbers (the sum of the first triangular numbers), the output corresponds to superfactorials (the product of the first factorials).
• If we consider square pyramidal numbers 1, 5, 14, 30, 55, 91, …, the output consists of determinants of specific n×n matrices defined in this context.

While I haven't yet proven these observations, I am eager to explore them further. This narrative simply needed to be shared because it is profoundly captivating.

Final Thoughts

I cannot fault the ancient Greeks for failing to find a proof for this theorem's most general case, as they did not even consider powers beyond cubes. They viewed numbers as representations of geometrical shapes and did not work in dimensions exceeding three. However, they might have discovered the case for n=3, where the partial sums correspond to the addition of certain "shells" on a cube.

I am excited to delve deeper into this area of study. I aim to discover a straightforward proof of Moessner’s theorem, although I have read that proving it is not as simple as it may appear.

If anyone discovers any intriguing outputs from Moessner’s procedure, I would be keen to hear about them.

This procedure defines a function between sequences, mapping natural numbers to the empty sequence, even numbers to squares, and so on. Keeping this in mind may prove helpful in studying it.

Thank you for joining me on this exploration.

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