I U B I R E A , F E R I C I R E A , C Ă U T A R E A // The Lost Notebook of Another Ramanujan

 IUBIREA ESTE TOT CEEA CE CONTEAZĂ

Love is all that matters  -  sună ca dracu'...

THE SEARCH

In the summer before starting my first year of university in the autumn, I found this problem in the magazine below (page 358).

"O : 71.  Let  $n\in\mathbb{N}^*$ . Show that there exist

  $r\in\mathbb{N}^*\;and\;a_1,\;a_2,\;\dots a_r,\;b_1,\;b_2,\;\dots b_r\in\mathbb{Q}$  such that

$\sqrt{n}=a_1\sin b_1\pi +a_2 \sin b_2\pi+\dots+a_r \sin b_r\pi.$

{Author : } Marcel ȚENA, teacher, Alexandria (Prize-winning problem at the SSM competition)"

          The author will subsequently publish a solution using the Kronecker-Weber theorem.

I saw something special in this problem, just as, later on, I saw something special in Alina.

Not long ago, I came across some scribbled pages containing an attempt to solve this problem.

LOVE

Among the pages I found, I also came across two letters that Christina, my future wife, had sent me.

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Eight days later, we got married in Sibiu.

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THE SEARCH - 2

          I picked up a notebook for scribbles, to which I gave a pompous title :

We start with the formulas from Exercise 5.

(b)  $\sqrt{n}=2^{n-1}\cdot \sin \frac{\pi}{2n}\cdot \sin \frac{2\pi}{2n}\cdot \dots \cdot \sin \frac{(n-1)\pi}{2n} \tag{1}$

Using the formulas for transforming trigonometric products into sums

 results in an expression :

$\sqrt{n}=a_1\cdot \sin \{or\;\cos\} b_1\pi+\dots +a_r\cdot \sin\{or\;\cos\}b_r\pi$

Whenever a cosine appears, it can be transformed into a sine using the complementary angle formula:

$\cos b\pi=\sin \left (\frac{\pi}{2}-b\pi\right )$

Then we finally obtain the desired writing of the  $\sqrt{n}$.

$\blacksquare$

LOVE  &  SEARCH

          The results obtained were surprising. We are including a few sheets showing the results and calculations.

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Wait, what is this strange paper with holes along the edge?

          This is the point I wanted to reach—the word *Felix*, which means happiness.

That’s how we worked back then—using Felix C-generation computers and dot-matrix printers... I was left with stacks of paper from the many projects I worked on during university (in Fortran, Assiris, and Cobol), and I used the backs of the sheets to sketch out calculations.

Ah, and now, another moment of nostalgia—this one quite recent. That is exactly how a conversation with Alina began—about "buffers" and logic diagrams...

< End Of Love>

THE SEARCH  3

          Especially in the case of prime numbers  $n$ , the formula seemed to have a distinctive structure.

The formula seemed linked to cyclotomic polynomials, as I had initially thought, but then certain Legendre symbols appeared to be involved...

     I thought I would hit the jackpot by going all the way to solving Fermat's Last Theorem. 

     Much later, I learned that Gauss had already obtained such formulas....

Next, I add all the old sheets I have left, without any particular order... And, indeed,

  THE SEARCH is sometimes more important than the result.

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THE END