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.

Thanks in advance.

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

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

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.

Regarding the second example, it is feasible to encourage students to find similar results of the same kind because there are many. They would be able experiment.

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    $\begingroup$ While this formula is indeed awe-inspiring to a mathematician, I suspect that most high-school students would find it less so. Not only would they not understand the meaning, they don't realize how few sums of this type have closed-form expressions. $\endgroup$ – Robert Israel Jan 13 '17 at 0:37
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    $\begingroup$ For another family of examples accessible to students, see also puzzle 12 at math.harvard.edu/~elkies/Misc/index.html#puzzles and its solution at math.harvard.edu/~elkies/Misc/sol12.html . $\endgroup$ – Noam D. Elkies Jan 13 '17 at 1:57
  • $\begingroup$ @T.Amdeberhan thank you for your reply! $\endgroup$ – Khadija Mbarki Jan 13 '17 at 6:32
  • $\begingroup$ @NoamD.Elkies Thank you for these references!! $\endgroup$ – Khadija Mbarki Jan 13 '17 at 6:33

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