Intro
Fibonacci pseudo-primes
Soliton recursion
Entropy
Graphene Cherenkov
Twisted light
Non-perturbative QED
Short times

- Our research sometimes brings us to face some beautiful pure-math challenges
- This page collects some of the math challenges that are representative of the theoretical work we have done,

and presents them as puzzles for the curious visitor - If you solved one of these puzzles and want to hear more about the research context and meaning,

dont hesitate to write or visit - If you solved 3 puzzles or more, I strongly encourage you to apply to my group

Interested in joining my research group at the Technion?

Fibonaci number of index $n$ modulo $n$ seems to give $\pm 1$ for every prime $n$ (except for $n=5$)

**Can you prove it?**

(There are non-primes that still satisfy this condition)

$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ |

$F(n)$ | $1$ | $1$ | $2$ | $3$ | $5$ | $8$ | $13$ | $21$ | $34$ | $55$ | $89$ | $144$ | $233$ | $377$ | $610$ | $987$ | $1597$ |

$F(n)\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n$ | $0$ | $1$ | $-1$ | $-1$ | $0$ | $2$ | $-1$ | $-3$ | $-2$ | $5$ | $1$ | $0$ | $-1$ | $-1$ | $-5$ | $-5$ | $-1$ |

Lucas numbers are much nicer than their Fibonacci relatives:

here it seems that $L(n)$ mod $n=1$, for every prime $n$, with no exceptions

$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ |

$L(n)$ | $1$ | $3$ | $4$ | $7$ | $11$ | $18$ | $29$ | $47$ | $76$ | $123$ | $199$ | $322$ | $521$ | $843$ | $1364$ | $2207$ | $3571$ |

$L(n)\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n$ | $0$ | $1$ | $1$ | $-1$ | $1$ | $0$ | $1$ | $-1$ | $4$ | $3$ | $1$ | $-2$ | $1$ | $3$ | $-1$ | $-1$ | $1$ |

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!**

${z}_{n+1}={F}_{c}({z}_{n},{w}_{n})$

${w}_{n+1}={F}_{d}({w}_{n},{z}_{n})$
$${F}_{g}(u,v):=\frac{(1-g+|v{|}^{2})u+gv}{g\stackrel{\u2015}{v}u+1+(1-g)|v{|}^{2}}$$

The parameter $c$ is an arbitrary complex number (imaginary part of zero typically gives degenerate dynamics),

and $d=f(c)$, with $f(c):=1-c+c/\stackrel{\u2015}{c}$

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 ${w}_{n}$?
**

$$F(X):={\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\frac{dy}{((1-{e}^{-y})X-y{)}^{2}+{\pi}^{2}}$$
**Prove that $F$ satisfies the following equation for all $X>0$**

$$\frac{1}{F(X)}-1+\mathrm{log}(\frac{1}{F(X)}-1)=\mathrm{log}(X)-X$$
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
$$(1+\rho )X=\rho \mathrm{log}(\rho )\phantom{\rule{2cm}{0ex}}\rho =G(X)$$
**What can you say about $X<0$? $\phantom{\rule{2cm}{0ex}}$ 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 $|a|<1$
$${\int}_{0}^{\pi}\frac{{\mathrm{cos}}^{2}(\theta )\mathrm{sin}(\theta )}{\mathrm{cos}(\theta )-a}d\theta $$
$${\int}_{0}^{\pi}\frac{{\mathrm{cos}}^{3}(\theta )}{\mathrm{cos}(\theta )-a}d\theta $$
$${\int}_{0}^{\pi}\frac{{\mathrm{cos}}^{2}(\theta )}{\mathrm{cos}(\theta )-a}d\theta $$

**Simplify the following integral**

$${\int}_{0}^{2\pi}d\varphi {e}^{ikx\mathrm{cos}(\varphi )+iky\mathrm{sin}(\varphi )}$$
**More generally, what function is the following integral representing for an integer $m$?**

$${\int}_{0}^{2\pi}d\varphi {e}^{-im\varphi}{e}^{ikx\mathrm{cos}(\varphi )+iky\mathrm{sin}(\varphi )}$$

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 ${\mathrm{\Gamma}}_{n}$ and ${G}_{n}$,

which are defined over domains of size $M$ and are zero everywhere else)
$${P}_{0}(t)=|\sum _{x}^{}{e}^{ixt}{G}_{0}(x){|}^{2}$$
$${P}_{1}(t)=\sum _{z1}{\mathrm{\Gamma}}_{1}({z}_{1})|\sum _{x}^{}{e}^{ixt}{G}_{0}(x){G}_{1}(x-{z}_{1}){|}^{2}$$
$${P}_{2}(t)=\sum _{z1}{\mathrm{\Gamma}}_{1}({z}_{1})\sum _{{z}_{2}}{\mathrm{\Gamma}}_{2}({z}_{2})|\sum _{x}^{}{e}^{ixt}{G}_{0}(x){G}_{1}(x-{z}_{1}){G}_{2}(x-{z}_{1}-{z}_{2}){|}^{2}$$
$${P}_{n}(t)=\sum _{z1}{\mathrm{\Gamma}}_{1}({z}_{1})\sum _{{z}_{2}}{\mathrm{\Gamma}}_{2}({z}_{2})\x85\sum _{{z}_{n}}{\mathrm{\Gamma}}_{n}({z}_{n})|\sum _{x}^{}{e}^{ixt}{G}_{0}(x){G}_{1}(x-{z}_{1}){G}_{2}(x-{z}_{1}-{z}_{2})\x85{G}_{n}(x-{z}_{1}-{z}_{2}-\x85-{z}_{n}){|}^{2}$$

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$({M}^{3}{n}^{3}\mathrm{log}(MN))$

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 $t$?**

$$U(t)={\int}_{e}^{\mathrm{\infty}}\frac{{e}^{i\omega t}}{\omega {\mathrm{log}}^{2}(\omega )}d\omega $$