# Eikonal Blog

## 2010.07.13

### Diferentiating primes

Filed under: mathematics — Tags: , — sandokan65 @ 15:50

Source:

• How to Differentiate a Number; by Victor Ufnarovski and Bo Ahlander; Journal of Integer Sequences, Vol 6 (2003), Article 0334
• T24661

Definition: On the set of integer numbers ${\Bbb N}_0=\{0,1,2,\cdots\}$ introduce following unary operation:

• $0' = 0,$
• $1' = 0,$
• $p' = 1, \ \ \ (\forall p \in {\Bbb P}),$
• $(ab)' = a'b+a b' \ \ \ \ (\forall a, b \in {\Bbb N}) \ \ (\hbox{Leibnitz rule}).$

Examples: $\begin{tabular}{ | c | c c c c c c c c c c c c c c c c c c |} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline n' & 0 & 1 & 1 & 4 & 1 & 5 & 1 & 12 & 6 & 7 & 1 & 16 & 1 & 9 & 8 & 32 & 1 & 81 \\ \hline n'' & 0 & 0 & 0 & 4 & 0 & 1 & 0 & 16 & 5 & 1 & 0 & 32 & 0 & 6 & 12 & 80 & 0 & 10 \\ \hline n''' & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 32 & 1 & 0 & 0 & 80 & 0 & 5 & 16 & 176 & 0 & 7 \\ \hline n'''' & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 80 & 0 & 0 & 0 & 176 & 0 & 0 & 32 & 368 & 0 & 0 \\ \hline \end{tabular}$

Note:

• $(a+b)' \ne a' + b',$
• $(a b)'' \ne a''b+2a'b'+a b''.$

Examples:

• $(2k)' = k + 2 k' \ge 2k',$
• $(3k)' = k+3k' \ge 3k',$
• $(4k)' = 4k + 4k' \ge 4k',$
• $(5k)' = k + 5k' \le (2k)' + (3k)'.$

Note: The only pairs $(a,b)$ s/t $a\le b \le 100$ with $gcd(a,b)=1$ and $(a+b)'=a'+b'$ are: $\{(1,2), (4,35), (4, 91), (8, 85), (11,11), (18, 67), (26,29), (27, 55), (35, 55), (36, 81), (38, 47), (38, 83), (50, 79), (62, 83), (95,99)\}$.

Theorem: $n'\ge n \Rightarrow (kn)' \ge kn, \ \ \ \ (\forall k > 1).$

Theorem: If $n = \prod_{i=1}^k {p_i}^{n_i}$ then $n' = n \sum_{i=1}^{k} \frac{n_i}{p_i}.$

Theorem: For $n = m p^p$, ( $p\in {\Bbb P}$, $m> 1, \in {\Bbb N}$)

• $n' = p^p (m+m'),$
• $\lim_{k\rightarrow \infty} n^{(k)} = \infty.$

## Extensions:

One can extend the differentiation operation to wider sets of numbers. For example, by setting $(-1)’=0$ one gets extension to the whole set of integer numbers ${\Bbb Z}$:

• $(-1)' = 0,$,
• $(-p)' = -1, \ \ \ (\forall p \in {\Bbb P})$,
• $(-x)' = - x', \ \ \ (\forall x \in {\Bbb Z})$.

One can also infer that:

• $i' = 0,$
• $\omega' = 0, \ \ \ (\forall \omega^m = 1, \ \ m\in {\Bbb N}),$
• $\omega' = 0, \ \ \ (\forall \omega^\alpha = 1, \ \ \alpha\in {\Bbb C}\ \{0\}).$
• $\left(\sqrt{p}\right)' = \frac{\sqrt{p}}{2p},$
• $\left(\frac1{p}\right)' = -\frac{1}{p^2},$
• $\left(p^\alpha\right)' = -\alpha p^{\alpha-1},$
• $\ln(p)' = \frac1{p},$
• $\left(\frac{n}{m}\right)' = \frac{n'm- n m'}{m^2}.$

Definition:

• $\pi(n) :\equiv \ (\# \ of \ p \le n),$
• $\pi_1(n) :\equiv \pi(n) - \pi(n-1).$

Lemma:

• $\pi_1(n) = 1$ if $n=p,$
• $\pi_1(n) = 0$ if $n\ne p.$

Lemma:

• $\pi_1(n) = 1 \Leftrightarrow n'=1,$
• $\pi_1(n) = 0 \Leftrightarrow n'\ne 1.$

## 2010.02.09

### Products

• $\prod_{n=1}^\infty \frac{(2n)^2}{(2n-1)(2n+1)} = \frac\pi2$, (John Wallis; 1616-1703)
• $\prod_{n=1}^\infty \frac{(4n)^2}{(4n-1)(4n+1)} = \frac{\pi\sqrt{2}}4$,
• $\prod_{n=1}^\infty \frac{(4n+2)^2}{(4n+1)(4n+31)} = \sqrt{2}$,
• Question: what is the value of $P_m:\equiv \prod_{n=1}^\infty \frac{(mn)^2}{(mn-1)(mn+1)}$? Clearly $P_2= \frac\pi2$ and $P_4= \frac{\pi\sqrt{2}}4$.
• $\prod_{n=1}^\infty \left(1+\frac{(-)^{n+1}}{2n-1}\right) = \sqrt{2}$,
• $\prod_{n=1}^\infty \left(1-\frac1{n^2}\right) = \frac12$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^2}\right) = \frac{\sinh(\pi)}\pi = 3.0\cdots$,
• $x \prod_{n=1}^\infty \left(1- \frac{x^2}{n^2\pi^2}\right) = \sin(x)$,
• $x \prod_{n=1}^\infty \left(1+ \frac{x^2}{n^2\pi^2}\right) = \sinh(x)$,
• $\prod_{n=1}^\infty \left(1- \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cos(x)$,
• $\prod_{n=1}^\infty \left(1+ \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cosh(x)$,
• $\prod_{n=3}^\infty \left(1-\frac4{n^2}\right) = \frac16$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^3}\right) = \frac{\cosh(\\pi\sqrt{3}/2)}\pi = 2.428,\cdots$,
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^s}\right) = \frac1{\zeta(s)}$ (Leonard Euler), where ${\Bbb P}$ is the set of prime numbers, and $\zeta(s):\equiv\sum_{n=1}^\infty \frac1{n^s}$ is the Riemann’s zeta function.
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^2}\right) = \frac6{\pi^2}$.

Strange forms:

• $\frac{\sqrt{2}}2 \frac{\sqrt{2+\sqrt{2}}}2 \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}2 \cdots = \frac{2}{\pi}$, (Francois Viete; 1540-1603).
• $\sqrt{x \cdot \sqrt{x\cdot \sqrt{x \cdots}}} = x$,
• $\sqrt[n+1]{x \cdot \sqrt[n+1]{x \cdots}} = \sqrt[n]{x}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} \frac1n & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & e \\ 0 & 1\end{pmatrix}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} 1 & \frac1{n!} \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 1 & e \\ 0 & 1\end{pmatrix}$, (which is quite trivial).

Sources:

• Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing