Interesting Numbers — Part 1
In this post, we will be looking at some numbers in mathematics that have some interesting properties and also have fun and imaginative names. Though they might not have much use in general in real life, it is still fun to look at them. This is the first part of this series.
Let us start with a very famous number.
Ramanujan Number
1729, one of the most famously known numbers, is also called the Ramanujan number or the Taxicab number. As to why it is called so, there is an interesting story behind it:
Once in London, when Ramanujan was ill and in hospital, fellow mathematician G.H.Hardy had engaged a cab with the number 1729 to visit him. On arriving, he remarked it was “rather a dull number” and hoped it was not a bad omen.
“No, Hardy,” said Ramanujan. “ It is a very interesting number. It is the smallest number expressible as the sum of two different cubes in two different ways”
1729 = 123+13= 93+103
Numbers of this form are called taxicab numbers. Other examples are:
Perfect Numbers
Perfect Numbers are numbers which are equal to the sum of their factors (excluding themselves). The smallest known perfect number is 6. Its factors are 1, 2, 3(1+2+3 = 6). The next one is 28, whose factors are 1, 2, 4, 7, 14(1+2+4+7+14 = 28)
Perfect numbers are very rare to find (Given that they are “perfect”). The next one is 496, 4th one is 8128 and the 5th perfect number is a staggering 33,550,336(Brave hearted souls can try to list out the factors and add them up!!)
Strangely, every perfect number found till now has been an even number 🤔🤔
Armstrong Numbers
Narcissistic Numbers, also called Armstrong Numbers (after Michael F. Armstrong), are numbers who are equal to the sum of their digits raised to the power of the number of digits. This is a very common question given in basic C/C++ programming classes.
A very simple and obvious example is 1. A few examples of 3-digit numbers are:
Other examples are 371, 407, 8208, 9474, which would be a good exercise to try out on your own.
There are some interesting twists within the Armstrong numbers.
- Dudeney Numbers: Add the digits of the number before raising them to the power of 3. eg: 5832 = (5+8+3+2)3, 512, 4917
- Munchausen Numbers: Raise each digit to the power of itself and add them up. eg: 3435 = 33+44+33+55
Pretty interesting right. And some people tell that mathematics is boring!!
Friedman Numbers
Adding on from Armstrong Numbers, the most interesting and fun of all, are the Friedman Numbers. They are named after Erich Friedman, a retired mathematics professor at Stetson University, Florida
Friedman numbers are numbers that can be written in a non-trivial way using their digits with the basic operations of arithmetic. A simple few examples:
There are certain Friedman numbers that are “nice” Friedman numbers (No. I did not make that term up. You can check it up online!!). These are numbers where the digits in the expression can be arranged in the same order as in the number itself.
- 216, 1024, 1255, 1260 are other examples of Friedman numbers
- 2187, 2592, 4096 are other examples of nice Friedman numbers
One interesting activity you can do is to see for yourselves whether the above are indeed Friedman numbers by trying to write them in the format as specified above. Do comment/let me know your answer.
I would like to end this post with a very interesting Friedman number:
999999999 = ((9+9+9)9–9 + 9)9–9/9
Will write about more such interesting numbers in the next part(s). Do let me know if you want me to write about any other interesting numbers you have come across. You can drop them in the comments or get in touch with me here.
Thanks for reading!!!
References
Originally published at http://infinitesimallysmallcom.wordpress.com on August 23, 2020.
