How I chanced upon Factorial Function

Today we will see how Piyush Goel(the contributor of this article) discovered something while playing with numbers. Here is the story contributed by him:

One day while sitting idle, and having nothing important to do I was just scribbling on paper. I was writing random numbers and thinking how I could link them. To my amazement I discovered this:

I wrote down 0, 1, 2, 3, 4, 5.

Next to each number I wrote their respective squares, viz. 0,1, 4, 9, 16, 25.

Then start subtracting each successive square from the next bigger square.

It looks something like this: (1-0), (4-1), (9-4), (16-9) and (25-16).

What is the result?

I got 1, 3, 5, 7, 9.

Now again subtract each successive number from the next one in the order.

We are performing this: (3-1), (5-3), (7-5) and (9-7).

Surprisingly, I got (2, 2, 2, 2). A 2 in each case. I was amazed and decided to take a step further. This time with cubes.

So here go my numbers 0, 1, 2, 3 ,4, 5, 6.

And their respective cubes: 0, 1, 8, 27, 64, 125, 216.

Time to perform first round of successive subtraction, i.e. (1-0), (8-1), (27-8), (64-27), (125-64) and (216-125). The result: 1, 7, 19, 37, 61, 91.

Second round of successive subtraction as performed earlier: (7-1), (19-7), (37-19), (61-37) and (91-61).

And this is what I got: 6, 12, 18, 24, 30.

Successive subtraction performed the third time: (12-6), (18-12), (24-18) and (30-24).

And this time I got the result (6, 6, 6, 6).

You see, squaring means ‘to the power 2’ and I got the result 2 after 2 successive subtractions, which is the value of 2!

Cubing means ‘to the power 3’ and after performing 3 successive subtraction I got 6 in each case which is nothing but the value of 3!

I similarly performed with 4th and 5th powers of the first few whole numbers. I was surprised to find the final value of 24 and 120 which are of course the values of 4! and 5!.

It was all right up to 5, but I wanted to go a notch higher and check for 6 just for my personal satisfaction. While doing for 6th power the calculations were massive and numbers so huge that it took me a couple of hours to do my calculations. Just to be sure I also did it for numbers upto 15.

Put 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 and their respective 6th powers:

0,1,64,729, 4096, 15625, 46356, 11649, 262144, 531441, 1000000, 1771561, 2985984,4826809, 7829536 and 11390625.

I kept on doing successive subtractions till to my contentment I finally got 720 for all which is the value of 6!

With little efforts, we can work out algebraic relationship for the above but I’ll leave it for some other time. Do share your comments.

About The Author:

Piyush Goel, born on 10th February, 1967. He has a Diploma in Mechanical Engineering and a Diploma in Vastu Shastra. Always wanting to do something new, Mr. Goel has written 15 Spiritual and World Famous Books with his own hands in Mirror Image in different ways. He is now known as “Mirror Image Man”. He also has World’s First Hand-written Needle Book “Madhushala” to his credit.

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Preeti Patawari

I'm here to share my love for numbers specially where I can help you clear your aptitude examinations like CAT, GMAT, IBPS and so on. In past I did that full time at Career Launcher.

3 thoughts to “How I chanced upon Factorial Function”

  1. Hello Vineet Patawari
    What an article, Wonderful work done by author by now I have not seen such an article on Factorial Function, I think it the existence of Factorial Function and what hard work done by the Author Heads off .
    Really Gr8

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