The New Year 2025: 2025 is an Amazing Number

My learned friend Prab Bhatt sent me some amazing properties of the number 2025 - the new year that has just started. I share these with some additions:

2025 is a perfect square:  2025 = 45 x 45

In our times, years that are perfect squares happen about once a century.  The previous square year was 1936, and the next will be 2116 - 91 years later.  

Notice that 45 is the sum of the two parts 20 and 25 in 2025.

(Other numbers that show this property of being chopped in  equal parts, and the parts added and squared to yield the same number are 81, 3025, 9801 - try them!)

2025 is the sum of three squares:  we can write 2025 as follows

                    2025 =  5² + 20² + 40² 

2025 is the product of two squares:  we can write 2025 as follows

                    2025 = 9² x 5² = 3² x 15²

2025 is the sum of cubes of all the digits from 1 to 9:  we can write 2025 as follows

The next year that is the sum of cubes of digits from 1 to 10 will be 3025 - 1000 years later.  Previously, 1296 was the year that was the sum of cubes of digits from 1 to 8!

2025 is the square of the sum of digits from 1 to 9

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9)²

2025 is interesting:  The sum of digits of 2025 is equal to 9.  

                               2 + 0 + 2 + 5 = 9

We can divide 2025 by 3 and the sum of digits of the quotient is again equal to 9. 

2025  ÷ 3 = 675.  Sum of digits of 675 is 6 + 7 + 5 = 18 

                              and the sum of digits in 18 is 9

And again:  675  ÷ 3 = 225.  Sum of digits of 225 is 2 + 2 + 5 = 9 

2025 and number 7:  If we break number 7 in its odd and even parts we have 

                       7 = 6 +1 = 4 + 3  = 2 + 5.  

The product of odd parts raised to the corresponding even part gives

Happy New Year!

Part 2: A Generating Function for Varg Fractions; Developing Generalized Varg Fractions

Recently, I had published a blog entitled 'Interesting Properties of Some Pythagorean Triples - Introducing Varg Fractions'  where I had introduced a class of fractions with some interesting properties.  

This article (Part 1) may be accessed here. In Part 1, we had used Pythagorean triples to generate Varg Fractions.  

What are Varg Fractions?  A varg fraction F is such that F+1 and F-1 are perfect squares. This implies that F²-1 = (F+1)(F-1) is also a perfect square. 

(If F <1, then 1+F, 1-F and 1-F² are perfect squares.)

As in Part 1, we shall only consider irreducible fractions such that the numerator (N) and the denominator (D) have no common divisor.  Also, since F+1 and F-1 are perfect squares, the denominator D of a varg fraction F must also be a perfect square.  The following slide (Slide 3 of Part 1) lists the first few varg fractions.

We can construct varg fractions using Pythagorean Triples, however, the method has some serious limitations (will become clear later in this article).  Here, I discuss a generating function for varg fractions that is easier to use and also allows generalisation of varg fractions that is not possible with Pythagorean triples.

Generating Function for F>1:  since square roots of F+1 and F-1 are both rational numbers, their sum is also a rational number (a rational number may be expressed as the ratio of two integers i.e. as a fraction).  

Slide 1 explains the details:

For Part 1, Click here.

Interesting Properties of Some Pythagorean Triples - Introducing Varg Fractions (Part 1)

Category:  Self-Indulgence

 please click on a slide to view its full screen version

In this publication, I have described some highly interesting properties of the varg (pronounced as v+erg) fractions (F). Varg means square in Hindi.  These may be generated from Pythagorean triples and I have discussed the process of doing so

  Part 2 may be accessed here.

Note: I am not a mathematician by profession, and it is possible that something similar has been published before.  Please let me know by sending an email to  ektalks@yahoo.co.uk if I have duplicated previous work. I shall duly acknowledge any previous research on this topic and suitably modify the text of my blog.  

In my next blog (Part 2) , I shall discuss a more general method, without using Pythagorean triples for generating varg fractions (F > 1).  

Talk 6: Our View of the World; Aging and Perception: Perceptions of the Brain Why Our Perception of Reality is Almost Always Wrong?

 Slides for previous talks may be viewed here:   

Talk1  Talk 2   Talk 3   Talk 4  Talk 5

The sixth and final meeting of the class consisted of two parts.  In the first hour we looked at how we view/understand the state of the world with the final hour devoted to a discussion of the effects of ageing on our perception & decision making, and a brief presentation on what we can do to help our brains stay healthy.  

For the first part, I have relied heavily on the outcome and analysis of many IPSOS surveys which are nicely presented in Bobby Duffy's book 'The Perils of Perception'.  In fact, in slide 1,  I introduce the books that I have found useful in preparing the course material.  Of course, there are many websites that provide excellent explanations on many aspects of the course covered in the six meeting.