How can one not start from Euclidean Geometry?

After the 19th century, a lot of Mathematicians came to the conclusion that Euclidean Geometry is “incomplete”. The reason was Euclid’s 5th axiom or as also called the parallel postulate, was not that obvious. After doubting this axiom, different geometries, non-Euclidean geometries, were born.

However, my point for this article is that I don’t quite understand why mathematicians called it “incomplete”. There’s no mathematics without postulates at all.

I can list some postulates that are true for me and I make my assumptions (not necessarily the rest of the world agrees), and based on my truth I can develop my theorems. There’s no problem in that, at all.

Euclid developed a geometry based on his axioms. Based on his truth, based on how he saw the plane or space. Mathematics might be sometimes nothing more than an agreement. We create some tools (axioms), the way we want, those tools can be anything, sometimes we might even need a dictionary to define the symbols we might use. That’s alright.

However, after we are done with that, the truth of mathematics will appear. The power of proofs will shine. One cannot call something incomplete because of the definitions or axioms. What should be questioned is actually the theorems and their proofs, if something there is incomplete.

If we doubt axioms, we start a new path of creating something new, not destroying an already developed work.

For me, Euclidean Geometry contributed a lot to my mathematical and creative thinking. At the beginning, it was easy to understand since the concepts made sense. In my High School years, I did Advanced Euclidean Geometry as well and it was one of the best decisions on where to invest my time.

I can certainly say that without doing Geometry at such a high level, I wouldn’t have been able to think the same way I do today. I wouldn’t have practiced the ability to move objects in my head or solving complicated problems using only basic theorems and definitions which usually took me 3 to 4 pages for the proof.

Do you know how to count?


How many different triads in positive integers equal 100? (no repetitions among the numbers of a triad),

a problem proposed from the twitter account @AfarBell.

 Is there an algorithm to count it? Or a specific pattern so there’s no need to write down every single triad?


The idea is to count in a right way, avoid repetitions and not to miss a triad! Particularly, in this problem the point is to have a fixed number and two boundaries, the imin and the imax where among them numbers are moved but at the same time they’re always equal. There is also a number, I named it, bubble number, due to the fact to avoid a triad of the form n,m,n, there’s a number that appears twice, so instead of letting that happen I put that number n in a “bubble” so it is inactive for a specific counting.

So, to avoid repetitions in my counting (like count a triad twice), I always start with a fixed number, for example, 1, then I count all the combinations of the triads that have the number 1 in it. So I actually break the problem into pieces. I have my fixed number, then I start with the following number 2, and so the last number of the triad will necessarily be 97. Then continue at 3 and so on. I stop at 49, the reason is that I always want my second number of the triad (imin) to be smaller than the third (imax), just to keep track of counting.

The bubble numbers appear only if the fixed number is even, due to the fact that there’s a triad of the form 2k, m, m, like 4, 48, 48, and we don’t want that triad. However, in order not to miss a number, I put it as a bubble number, so I don’t forget any of them.

Finally, at the imax, there’s a subtraction of 2 every time from the biggest number, and that is controlled from the first number of imin in order the sum always to be 100.

So eventually adding all the totals of every counting we get eventually 784 triads.


Fixed Number               imin                      imax                    total             Bubble number

1                                         2,…,49                   50,…,97                48                                          

2                                        3,…,48                    50,…,95                46                               49       

3                                       4,…,48                     49,…,93                45                                            

4                                       5,…,47                     49,…,91                43                               48

5                                       6,…,47                     48,…,89                42                                           

6                                       7,,46                     48,,87                40                               47

7                                       8,…,46                     47,…,85                39                                                  

8                                       9,…,45                      47,…,83               37                               46          

9                                      10,…,45                    46,…,81                36

10                                    11,…,44                    46,…,79                34                                45

11                                    12,…,44                    45,…,77                33                 

12                                    13,…,43                     45,…,75               31                                44

13                                    14,…,43                     44,…,73               30                     

14                                    15,…,42                     44,…,71               28                                43

15                                     16,…,42                     43,…,69              27            

16                                     17,…,41                     43,…,67              25                                 42

17                                     18,…,41                     42,…,65              24       

18                                    19,…,40                     42,…,63               23                                 41

19                                    20,…,40                     41,…,61                21                        

20                                    21,…,39                     41,…,59                19                                40

21                                    22,…,39                     40,…,57                 18

22                                    23,…,38                     40,…,55                 16                                 39

23                                    24,…,38                     39,…,53                 15           

24                                    25,…,37                     39,…,51                 13                                 38

25                                     26,…,37                     38,…,49                12

26                                    27,…,36                     38,…,47                 10                                 37

27                                    28,…,36                     37,…,45                   9                    

28                                     29,…,35                     37,…,43                  7                                   36

29                                     30,…,35                     36,…,41                  6

30                                     31,…,34                     36,…,39                  4                                   35

31                                   32,…,34                     35,…,37                    3

32                                        33                               35                          1                                  34

The importance for students to know “why”

I really enjoyed my fist year in college, however, there was something that made me feel bad about myself, that I was not good enough. All of my classes had something in common; every professor assumed that the students are familiar with the concepts of the corresponding class. Even if sometimes I was already familiar with them I always was wondering “Why are we supposed to know that?”.

Spring of 2017, I took a class about Differential Equations. The first four weeks we had an introduction to Linear Algebra since we would need it later.

Eventually, two or three weeks before finals, we were doing Laplace Transform. I was sure I’ve had heard that name before and I was excited to get started.

My professor started with the definition:

CodeCogsEqn (8)

I was like “huh, so that’s Laplace transform”.

I remember him evaluating some Laplace transforms, starting with L{1}.
I asked one of my classmates “Did I miss any lecture or is it the first day we’re doing Laplace?”. It was indeed the first time we’re doing Laplace transforms.

I was waiting to understand what was going on there, and why we were doing Laplace transform anyway, my professor kept continuing on examples.

I couldn’t keep it inside me anymore. I raised my hand, he asked me if he missed anything again “No, actually I have a question this time… Why are we doing Laplace transform?” – I asked.

He laughed and told me that it is a good question, “Laplace transform is going to be later a useful tool to solve Differential Equations with real coefficients ” he eventually answered.

I was thinking at the moment ” Why he didn’t tell us anything at the beginning of the class, how are we supposed to know why we are doing what we are doing?”.

Unfortunately, I cannot interrupt every single lecture to ask “why” or “where are we going to need it”. I guess it is every professors’ responsibility to let students know at the beginning of the lecture if needed. Sometimes even if I am a math major and understand usually everything that we are doing in my math classes, but the answer to the question why it’s not always obvious.



The beauty of Diophantine equations

A Diophantine equation is usually an equation with two or more unknown variables, where their domain is the set of the integer numbers or a narrower set.

When I was in high school, I loved dealing with this kind of equations and there is a reason for that. In Diophantine equations, there’s no obvious way in how to even start. You can guess some numbers and if you’re lucky maybe those numbers satisfy the equality. However, that’s not enough. By guessing a solution, that doesn’t mean that it is unique.

The following problems are my 4 favorite problems from my notes of 100 problems collection in Diophantine equations:
CodeCogsEqn (4)


CodeCogsEqn (5)


CodeCogsEqn (3)


CodeCogsEqn (2) - Copy

The first three problems are from the book “An Introduction to Diophantine Equations” of T. Andreescu, D. Andrica, and I. Cucurezeanu.

And the last one is from the book “Problem-Solving Strategies” of A. Engel.


Response to a Developer about Euclid’s proof


This post is a response to some doubts about Euclid’s proof of prime numbers from “Notes of a Developer “.

So, before you even start reading this you may want to take a look at this post:

Bringof’s approach on how Euclid’s proof might be incomplete is a really good observation. Especially, at some point, he wrote that there is a correct and a false version which are, according to him:


It follows a numerical example in order to explain his statement and correction:


However, the false is the correct one, and the correct is false.


The numerical example shows that the number 30031 is not prime, that’s true. The above-mentioned example would prove wrong the following statement: “Every number of the form p1*p2*p3*…*pn + 1 is a prime number, where { p1, p2, p3,…,pn } are prime numbers “. It is not what Euclid said.

Euclid said that:

Let suppose that there is a finite set of prime numbers and let that be { p1, p2, p3,…,pn}. That means that in a perfect world those are the only prime numbers. There are no others. He continued his constructive proof by saying that there is a number m that m=p1*p2*p3*…*pn + 1.

But none of the prime numbers we know can divide m. With the assumption he made, that means that m is a prime number itself and doesn’t belong to the set of the prime numbers which is Contradiction!

Do we care if m is actually a prime or not? No, because with our assumption it is a prime number.

Going back to the numerical example the assumption is that: the only prime numbers that exist are { 2, 3, 5, 7, 11, 13 }, none of these numbers can divide 30031, that makes 30031 a prime number which doesn’t belong to the set of the prime number, contradiction!

With the assumption we made we don’t actually know that 59 or 509 are prime numbers. {2, 3, 5, 7, 11, 13} are the only prime numbers that exist according to the assumption we made.

Euclid’s proof of prime numbers

I was 15 when I first learned about this proof. The Mathematician, who was a volunteering math trainer to me, was the one who first introduced that proof to me.

He was talking about that beauty, how in math we can agree with someone “ Okay, let suppose that you are right” and then prove them wrong.

It’s amazing how powerful and beautiful the method of “proof by contradiction” is.

In addition, this proof became my favorite.


In Math, every single word is significant

Even if I’m studying Applied Mathematics, I always had a great interest in Number Theory, it was so “simple” and “fun” for me (at least in high school -but in general it’s not that simple at all).

Since I’m talking about Number Theory, I’m going to talk about the Prime Numbers, they are my favorites, they’re so “unique”.


When I was in the first grade of high school, I first learned about the above-mentioned theorem. I also used it a couple of times to solve/prove some problems, it was a useful one.

However, we all know that for the prime numbers, there’s no formula or function that will give you any p prime number by plugging in an i number (inN) and p will be the i-th prime number.

One day, I was thinking about that theorem, but in a quite different way ( in a wrong way). I didn’t remember exactly the theorem at the time and I had it in my head as that “any number of the form 6k+1 or 6k+5 is a prime number.”

So in my head “problem is solved”, “I can find any prime number by plugging in a number for k.”

It felt so perfect and easy to be true, I had that feeling that I usually have when I do something wrong or I have an incorrect assumption in Math.

It is easy to see that by plugging in some number for k will give you a non-prime number (for example k=15, it gives 91 and 95), I realized it the very following hour.

Then I was looking for that theorem, on a textbook, I had in my head and when I found out that I was totally off, I was like “yeah, you’re really smart.”

From that day, I pay more attention not to miss a single word in math and to remember every definition or theorem the exact way that is written in the textbook. In math, we don’t like to add words that are not important just to make it “sound” better. Everything is so precise, that’s the beauty of Math anyway.