Is 1729 really a boring number?

G.H Hardy,  a famous  number theorist, also mentored the Indian  mathematician Srinivasa Ramanujan.

Once, when Ramanujan was at a hospital in Putney, Hardy visited him. Hardy rode a taxicab numbered 1729 (came to be called as Hardy-Ramanujan number) on the way there, and he remarked that 1729 was a dull number. 1729 was really not boring at all!  Ramanujan instantly replied to Hardy that it was the smallest number to be written as a sum of two positive cubes in two different ways.
1729 = 1³ + 12³ = 10³+11³

But it also has a few other interesting properties!

• 1729=7*13*19, a product of three primes. Any number that is a product of three primes is called a Sphenic number.

• The three primes 7, 13, 19 satisfy 13=1*7+6  and  19=1*13+6. Such a number is a Zeisel number.

• 1+7+2+9=19, 19*91=1729.  This property was discovered by Masahiko Fujiwara.

Why is square root of 2 irrational?

In this and the next few blogs I will write proofs of irrationality that I enjoyed learning recently.  I will start with the simplest one on the proof of irrationality of square root of 2.

Have you ever wondered how many primes are there.

TA for Calculus BC class

I enjoy being a TA for the Calculus BC and AB class this year.  I never imagined that I will get to be a TA at 13.  I did the course for my AP Calc BC last year.