Real Mathematics – Special Numbers #1

Understanding Time

How did humans keep track of time before the invention of clock?

First thing comes to mind is that humans observed rise and fall of the Sun that is 1 day. People needed to know time in a broader aspect as civilizations emerged so that they could understand crucial time intervals such as rain season etcetera. For that, they used a cycle that was up there in the sky, changing its shape in a pattern: Phases of the moon. They observed and calculated that Moon phases are in cycles of 29 days.


A lunar month is roughly 29 days. There is about 1 day difference between lengths of a month and a lunar month.

29 is a whole integer.

If you try to divide it into two equal parts, it will give you 14,5 which is not a whole integer.

If you continue dividing 29 into equal whole integers, you would fail for every number between 2 and 28. Only there is one 29 and twenty nine 1s. Nothing else can divide 29 into equal whole integers.


In other words, there are only two whole numbers which gives 29 when they are multiplied: 1 and 29. These kinds of numbers are called “prime numbers”.

Today “one month” refers to 30 days. One of the main reasons behind turning 29 days into 30 is that 30 is not a prime number. It can be divided equally into whole numbers such as 2, 3, 5, 6 (2*3), 10 (2*5) and 15 (5*3).

Prime Numbers and Encrypting

Another beautiful property about 30 is that it can be shown as the multiplication of 2, 3 and 5. All these three numbers are prime. Therefore we can write any whole integer as the multiplication of prime numbers.

Let’s take this one step further: Every whole integer has one and only one representation what is a multiplication of prime numbers. Hence different integers have different prime number demonstrations.

This property has a key importance in our modern day. Assume that I have to create a password for my online banking account. Bank’s application requires me to create this password with using numbers only. Naturally I would like to create a password that would be very hard (preferably impossible) to crack. Also I would like to use an easy method when I create the password so that I can remember the method in case I forget the password itself.

Trapdoor Function: Roughly, it is a method that is easy to compute in one direction, yet difficult to compute in the opposite direction. You could imagine it as a machine that takes a number (input number) and easily processes it into another number (output number). Although it would be very hard or even impossible to use the output number and find the input number with it.

If I take one small and one huge (let’s say 6-digits) prime numbers, it would take only a few seconds to multiply them using a calculator. Resulting number, my new password, can only be found with the multiplication of those specific prime numbers. Then, if someone tries to log in to my bank account, he/she must know the primes I’ve chosen.

Creating 1115035 is very easy: Just multiply prime number 223007 with another prime 7.

Finding the prime multiplication of a random number is a really hard and long lasting thing to do.


Because, even today we don’t have a method that finds the prime multiplication of any given number. Thus, using prime numbers is an effective way to secure information.

Crack the Code

I’ve built an encryption system that uses prime numbers and alphabet. According to my system, every letter in the alphabet is assigned to a number:



  • I choose two prime numbers between 1 and 20.
  • I multiplied them.
  • Then I added the result to each letter in the alphabet.
  • For instance if I choose 3 and 5, letter A would go to (3*5=15, 15+1=16) the 16th letter that is the letter M. Hence letter M in my encrypt means that it is the letter A in reality.

Now, try to crack the encrypted sentence below:


M. Serkan Kalaycıoğlu

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