Adonis Diaries

The mathematical miracle that Euler started?

Note: I guess this article was truncated and could be more useful? Euler demonstrated that planets circles in ellipses around the sun, by calculation and by observations.

Gabriel Miranda

Oct 12, 2021

The most beautiful event I have ever seen was finding Euler, a name that appears many times all over mathematics. A 18th century mathematician that was so productive that even when he got blind he started working even harder

The amount of mathematical work out there made by this mathematician is astounding in it of itself. The work is so amazing and so profound and deep that it is crazy.

As a start we must ask ourselves a simple question: in how many ways can you write a number as a sum of other numbers without any repetition of a way changing it’s order?

This is quite a hard question to get around, but let’s consider finding these for a number, let’s say 4, now these so called “ways” will all be exactly: This is what is called the partitions of 4 because these are ways to partition 4 into other positive integers.

Now, let’s say that p(4) for example, represents the amount of partitions of 4, then p(4)=5.

A very absurd thing, believe it or not, is that the p(100)=190,569,292 which just means that there are 190,569,292 ways to write 100 as a sum of positive integers.

The amount of partitions of a number is a very deep mathematical question that Euler asked himself many times and he found something very peculiar.

One of the deepest truths that Euler discovered. A thing that can only be seen once in a lifetime being discovered, a truth so deep that stayed as one of the only ways to calculate partitions only until the genius of Ramanujan came.

A very simple, thing found by Euler that can be used to calculate the amount of partitions of a number exactly based on the amount of other partitions.

Let us try something very specific. Let’s first list some values of the amount of partitions of numbers.

p(0)=1 (is helpful)
p(1)=1
p(2)=2=p(1)+p(0)
p(3)=3=p(2)+p(1)
p(4)=5=p(3)+p(2)
p(5)=7=p(4)+p(3)-p(0)
p(6)=11=p(5)+p(4)-p(1)
p(7)=15=p(6)+p(5)-p(2)-p(1)
p(8)=22=p(7)+p(6)-p(3)-p(1)
p(9)=30=p(8)+p(7)-p(4)-p(2)
p(number)=p(number-1)+p(number-2)-p(number-5)-p(number-7)-…

Now what are these numbers, (1, 2, 5, 7, 12, 15, 22, …), that appear in these offsets?

Look at the numbers at every odd position, (say the first, the third, the fifth and so on), if we list them: 1, 5, 12, 22, …

Believe it or not there is a very simple definition for them, although they might just seem quite random, this definition is exactly that for the nth number of these we have: And believe it or not this is exactly what it is, and as for the even numbers, the exact value for the nth number of them is:

These numbers are what are called “pentagonal numbers” which comes from a very peculiar origin of which I am not going to show here, but, in the end the thing that comes out from all of Euler’s work, (and that he proves), is that for n being any number you choose, for p(0)=1 and for all negative numbers the amount of their partitions is 0