# Beauty of some numbers discovered by Ramanujan

I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and see the beauty of numbers. For instance, I have enough informations about the golden ratio. I watched Professor Ken Ono on national geographic speaking about the beauty of Pi discovered by Ramanujan. I also heared about the story of the taxi cab number $1729.$ Could you help me by providing me with some references or informations about some discovered facts about some special numbers that Ramanujan did. I wanted my pupils to see the beauty and the genuis in Ramanujan's work.

• There are many beautiful formulae arising from the problems Ramanujan submitted to the Indian Mathematical Society: see math.uiuc.edu/~berndt/jims.ps. – Mark Wildon Jan 13 '17 at 12:15

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.
In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.