Fibonaci number of index $n$ modulo $n$ seems to give $\pm1$ for every prime $n$ (except for $n=5$)
Can you prove it?
(There are non-primes that still satisfy this condition)
| $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ |
| $F(n)$ | $1$ | $1$ | $2$ | $3$ | $5$ | $8$ | $13$ | $21$ | $34$ | $55$ | $89$ | $144$ | $233$ | $377$ | $610$ | $987$ | $1597$ |
| $F(n) \mod n$ | $0$ | $1$ | $-1$ | $-1$ | $0$ | $2$ | $-1$ | $-3$ | $-2$ | $5$ | $1$ | $0$ | $-1$ | $-1$ | $-5$ | $-5$ | $-1$ |
| $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ |
| $L(n)$ | $1$ | $3$ | $4$ | $7$ | $11$ | $18$ | $29$ | $47$ | $76$ | $123$ | $199$ | $322$ | $521$ | $843$ | $1364$ | $2207$ | $3571$ |
| $L(n) \mod n$ | $0$ | $1$ | $1$ | $-1$ | $1$ | $0$ | $1$ | $-1$ | $4$ | $3$ | $1$ | $-2$ | $1$ | $3$ | $-1$ | $-1$ | $1$ |
The recursion relation below describes two sequences of complex numbers $z_n$ and $w_n$
What can you say about these sequences for all $n$? Try simulating it!
$z_{n+1} = F_{c}(z_n,w_n)$
$w_{n+1} = F_{d}(w_n,z_n)$
$$
F_g(u,v):=\frac{(1-g+|v|^2)u+gv}{g \bar v u + 1 + (1-g)|v|^2}
$$
The parameter $c$ is an arbitrary complex number (imaginary part of zero typically gives degenerate dynamics),
and $d=f(c)$, with $f(c):=1-c+c/ \bar c$
Note that $f$ has a nice symmetric property: $c=f(d)=f(f(c))$
Given an arbitrary choice of $z_0$, $w_0$, and $c$, what is the behavior of the sequences $z_n$ and $w_n$?
$$
F(X):=\int_{-\infty}^{\infty} \frac{dy}{\big((1-e^{-y})X-y \big)^2 + \pi^2}
$$
Prove that $F$ satisfies the following equation for all $X>0$
$$
\frac{1}{F(X)} - 1 + \log\Big(\frac{1}{F(X)} - 1\Big) = \log(X) - X
$$
Or, in other words, prove that $1/F(X) = 1 + X/G(X)$,
where the function $G$ is defined as the solution of the following
$$
(1+\rho)X=\rho \log(\rho) \hspace{2cm} \rho=G(X)
$$
What can you say about $X<0$? $\hspace{2cm}$ Any guess what is the physical importance of this integral?
Find, prove, or guess the correct analytical expressions for the following
for every real-valued $a$, and especially $|a|<1$
$$
\int_{0}^{\pi} \frac{\cos^2(\theta)\sin(\theta)}{\cos(\theta)-a} d\theta
$$
$$
\int_{0}^{\pi} \frac{\cos^3(\theta)}{\cos(\theta)-a} d\theta
$$
$$
\int_{0}^{\pi} \frac{\cos^2(\theta)}{\cos(\theta)-a} d\theta
$$
Simplify the following integral
$$
\int_{0}^{2\pi} d\phi e^{ikx \cos(\phi) + iky \sin(\phi)}
$$
More generally, what function is the following integral representing for an integer $m$?
$$
\int_{0}^{2\pi} d\phi e^{-im\phi}e^{ikx \cos(\phi) + iky \sin(\phi)}
$$
The goal is to find an efficient way to calculate $P_n(t)$ for large $n$ and an arbitrary t
($z_1,z_2,...,z_n$ and $x$ sum over all elements in the domains of the corresponding functions $\Gamma_n$ and $G_n$,
A naive calculation will have a time complexity O$(M^{n+1})$, which is inefficient since it grows exponentially in $n$
Find an efficient algorithm for calculating $P_n(t)$
Hint: can you find an algorithm that has a setup time of complexity O$(M^3n^3\log(MN))$
with additional O$(MnT)$ operations to calculate $P_n(t)$ for $T$ values of $t$?
What is the behavior of $|U(t)|$ at very small $t$?
$$
U(t) = \int_{e}^{\infty} \frac{e^{i \omega t}}{\omega \log^2(\omega)} d\omega
$$