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.
.
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.
.
.
.
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.
