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Beauties

This is a collection of mathematical results that I consider as gems based on their beauty and elegance alone. Neither the depth nor the width of its application is criteria for consideration in my collection. It is only the beauty that matters here. Hence the title ‘Mathematical Gems. Again as they say, beauty lies in simplicity, simplicity of the statement of the result is therefore one of my criteria. Also since number theory is my area of interest, a majority of my gems belong to number theory. For each gem, I have given a one line explanation why I saw a gem in it. But I believe that for these gems, the best explanation is in the theorem itself.

Gem 1An all in one formula

This asymptotic formula is one of my personal favorites because it unites most of the important topics and constants of analytical number theory under one single formula:

Apéry’s constant $\zeta(3)$,

Euler-Mascheroni constant $\gamma$,

Harmonic number $H_n$,

Merten’s constant $M$,

n-th prime numbe$p_n$,

non trivial zeros of the Riemann zeta function $\gamma_n$

and Pi $\pi$.

$\displaystyle{\sum_{r=1}^{n} \frac{1}{\gamma_r}\Big(\frac{1}{r}+\frac{1}{p_r}\Big) \Big(\tan^{-1} \frac{\gamma_r}{\gamma_n}\Big)^2e^{H_r + \frac{1}{p_1} + ... + \frac{1}{p_r}}\sim \frac{e^{\gamma + M}}{4}\Big(K - \frac{7\zeta(3)}{4\pi}\Big)\ln^2 n}$.

Gem 2. Fibonacci and Lucas numbers

This formula due to R C Johnson gives a beautiful relationship between the Fibonacci numbers and the Lucas numbers.

Define the Fibonacci number $F_n$ and the Lucas number $L_n$ as

$F_1 = F_2 = 1$ and, for $n > 2$, $F_{n} = F_{n-1} + F_{n-2}$ and

$L_1 = L$, $L_2=3$ and, for $n > 2$, $L_{n} = L_{n-1} + L_{n-2}$

respectively. Then,

$\exp (L_n+\frac{L_{2n}x^2}{2}+\frac{L_{3n}x^3}{3} + ...) =\frac{1}{F_n}(F_n+F_{2n}x+F_{3n}x^2 + ...)$

Gem 3. Maximum value of a monic polynomial

Ramachandra and Balasubramanian (and possibly Chebyshev)

If $f(x)$ is a polynomial having integral coefficients and the coefficient of the highest order term is 1 (monic polynomial) then, on the interval $-2 \le x \le 2$, $f(x)$ must attain a magnitude of at least 2: i.e.

$\max_{-2 \le x \le 2}|f(x)| \ge 2$.