Skip to main content

Ramanujans and Hardy Partition Function

 Hello Aliens, This blog is about the very famous Partition theory of numbers by Srinivasa Ramanajun and GH hardy. 




The theory of partitions of numbers is an interesting branch of number theory. The concept of partitions was given by Leonard Euler in the 18th century. After Euler though, the theory of partition had been studied and discussed by many other prominent mathematicians like Gauss, Jacobi, Schur, McMahon, and Andrews, etc. but the joint work of Ramanujan with Prof. G.H. Hardy made a revolutionary change in the field of partition theory of numbers. Ramanujan and Hardy invented the circle method which gave the first approximations of the partition of numbers beyond 200.


A partition of a positive integer ‘n’ is a non-increasing sequence of positive integers, called parts, whose sum equals n. In simpler words, it means the number of ways in which a given number can be expressed as a sum of positive integers. 

For example, p(4) = 5, i.e. there are five different ways that we can express the number 4. The partitions of the number 4 are:

4,

3+1,

2+2,

2+1+1,

1+1+1+1

Here we can express 4 in the above-mentioned different terms including the number itself.


Lets take another example say 5. Now we can express it as-

5

4 + 1

3 + 2

3 + 1 + 1

2 + 2 + 1

2 + 1 + 1 + 1

1 + 1 + 1 + 1 + 1

That means we can write 5 as the sum of positive integers in 7 ways. 


In general, we take p(0) as 1, which means there is only one way to represent zero as the sum of positive integers. Similarly, we can say for negative integers too.


We can write p(n)  as :


                    p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+............

 

We can find the partition of small numbers after bit calculation but as the number increases the partition increases more exponentially. For example, you might be surprised that 10 has 42 partitions while 100 has more than 190 million partitions. Isn't it amazing? Both Ramanujan and Hardy derived an approximation asymptotic formula for the number of partitions. It is -









Srinivasa Ramanujan is credited with discovering that the partition function has nontrivial patterns in modular arithmetic. For example, the number of partitions is divisible by five whenever the decimal representation of ends in the digit 4 or 9, as expressed by the congruence.



For example, the number of partitions for the integer 4 is 5 that is divisible by five. For integer 9, the number of partitions is 30, which is divisible by 5 and for 14 there are 135 partitions, which is divisible by 5 too.

Ramanujan also discovered congruences modulo 7 and 11.




 The study of Ramanujan type congruence is an interesting and popular research topic of number theory. Because of its great applications in different areas like probability and particle physics (especially in quantum field theory), the theory of partitions has become one of the richest research areas of mathematics in recent times. More systematic study and a better understanding of partition theory will surely help in the advancement of mathematics with a new dimension.

Hope you have learned something new today. If you do so, then share this with others also. Till then have a good day.

 





Comments

Post a Comment

Popular posts from this blog

How to set up your Alexa account ?

Technology is controlling the world and one such technology is Artificial intelligence. Now we can see AI-based digital products in every house.  There's no lack of digital assistants like- Siri, Cortana, Bixby, Google Assistant, etc.  But users choose Alexa. A question must arise in your mind that why Alexa is the top preference of customers?  Is it because of its amazing technology and performance?  Amazon Echo  is a standalone speaker that needs nothing more than a power connection.  This blog is all about how to set up our Alexa account.  The first step is to install the Amazon Alexa App on your device. You can get it free on iOS, Android, or Fire OS. You just need an Amazon account to continue using it and Amazon Prime.   Once you sign in to the app you are good to go to the second step. The Second Step is to Pick a device to set up Alexa.  a) Click on the hamburger menu on the lower right and then select the 'Add a device' option presen...

7 years of teaching experience

 It was the year 2013 when we shifted from a small village to a big city.  I saw huge buildings for the first time. It was the first time I saw this huge number of people in one place. The first time, I got to know what a private school actually looks like. 🤯 I was new to this city culture, but it did not take much time to get used to it.  My mom, who used to be a house maker for 18 years, decided to give her a new career. She asked my father to create her resume and started giving interviews to private schools for the position of primary teacher in any school. Fortunately, she tried in the same school as mine, and she got the job. I remember, my mom told me what my school principal said to her, even if she does not have perfection in English, she got this job because she got inspired seeing her journey, and starting a new career after 18 years. She believed in my mom because of her confidence. She realized it on the first meet itself.  And from then, there is no go...