### Intro

• 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,
don’t hesitate to write or visit

• If you solved 3 puzzles or more, I strongly encourage you to apply to my group

### Fibonacci pseudo-primes

Fibonaci number of index $n$$n$ modulo $n$$n$ seems to give $±1$$\pm1$ for every prime $n$$n$ (except for $n=5$$n=5$)
Can you prove it?
(There are non-primes that still satisfy this condition)
 n$n$$n$ 1$1$$1$ 2$2$$2$ 3$3$$3$ 4$4$$4$ 5$5$$5$ 6$6$$6$ 7$7$$7$ 8$8$$8$ 9$9$$9$ 10$10$$10$ 11$11$$11$ 12$12$$12$ 13$13$$13$ 14$14$$14$ 15$15$$15$ 16$16$$16$ 17$17$$17$ F(n)$F\left(n\right)$$F(n)$ 1$1$$1$ 1$1$$1$ 2$2$$2$ 3$3$$3$ 5$5$$5$ 8$8$$8$ 13$13$$13$ 21$21$$21$ 34$34$$34$ 55$55$$55$ 89$89$$89$ 144$144$$144$ 233$233$$233$ 377$377$$377$ 610$610$$610$ 987$987$$987$ 1597$1597$$1597$ F(n)modn$F\left(n\right)\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n$$F(n) \mod n$ 0$0$$0$ 1$1$$1$ −1$-1$$-1$ −1$-1$$-1$ 0$0$$0$ 2$2$$2$ −1$-1$$-1$ −3$-3$$-3$ −2$-2$$-2$ 5$5$$5$ 1$1$$1$ 0$0$$0$ −1$-1$$-1$ −1$-1$$-1$ −5$-5$$-5$ −5$-5$$-5$ −1$-1$$-1$

Lucas numbers are much nicer than their Fibonacci relatives:
here it seems that $L\left(n\right)$$L(n)$ mod $n=1$$n=1$, for every prime $n$$n$, with no exceptions
Can you prove it?
are there non-primes that still satisfy this condition?
 n$n$$n$ 1$1$$1$ 2$2$$2$ 3$3$$3$ 4$4$$4$ 5$5$$5$ 6$6$$6$ 7$7$$7$ 8$8$$8$ 9$9$$9$ 10$10$$10$ 11$11$$11$ 12$12$$12$ 13$13$$13$ 14$14$$14$ 15$15$$15$ 16$16$$16$ 17$17$$17$ L(n)$L\left(n\right)$$L(n)$ 1$1$$1$ 3$3$$3$ 4$4$$4$ 7$7$$7$ 11$11$$11$ 18$18$$18$ 29$29$$29$ 47$47$$47$ 76$76$$76$ 123$123$$123$ 199$199$$199$ 322$322$$322$ 521$521$$521$ 843$843$$843$ 1364$1364$$1364$ 2207$2207$$2207$ 3571$3571$$3571$ L(n)modn$L\left(n\right)\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}n$$L(n) \mod n$ 0$0$$0$ 1$1$$1$ 1$1$$1$ −1$-1$$-1$ 1$1$$1$ 0$0$$0$ 1$1$$1$ −1$-1$$-1$ 4$4$$4$ 3$3$$3$ 1$1$$1$ −2$-2$$-2$ 1$1$$1$ 3$3$$3$ −1$-1$$-1$ −1$-1$$-1$ 1$1$$1$

### Soliton recursion

The recursion relation below describes two sequences of complex numbers ${z}_{n}$$z_n$ and ${w}_{n}$$w_n$
What can you say about these sequences for all $n$$n$? Try simulating it!

${z}_{n+1}={F}_{c}\left({z}_{n},{w}_{n}\right)$$z_{n+1} = F_{c}(z_n,w_n)$
${w}_{n+1}={F}_{d}\left({w}_{n},{z}_{n}\right)$$w_{n+1} = F_{d}(w_n,z_n)$ ${F}_{g}\left(u,v\right):=\frac{\left(1-g+|v{|}^{2}\right)u+gv}{g\overline{v}u+1+\left(1-g\right)|v{|}^{2}}$
The parameter $c$$c$ is an arbitrary complex number (imaginary part of zero typically gives degenerate dynamics),
and $d=f\left(c\right)$$d=f(c)$, with $f\left(c\right):=1-c+c/\overline{c}$$f(c):=1-c+c/ \bar c$
Note that $f$$f$ has a nice symmetric property: $c=f\left(d\right)=f\left(f\left(c\right)\right)$$c=f(d)=f(f(c))$
Given an arbitrary choice of ${z}_{0}$$z_0$, ${w}_{0}$$w_0$, and $c$$c$, what is the behavior of the sequences ${z}_{n}$$z_n$ and ${w}_{n}$$w_n$?

### Entropy

$F\left(X\right):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dy}{\left(\left(1-{e}^{-y}\right)X-y{\right)}^{2}+{\pi }^{2}}$ Prove that $F$$F$ satisfies the following equation for all $X>0$$X>0$
$\frac{1}{F\left(X\right)}-1+\mathrm{log}\left(\frac{1}{F\left(X\right)}-1\right)=\mathrm{log}\left(X\right)-X$ Or, in other words, prove that $1/F\left(X\right)=1+X/G\left(X\right)$$1/F(X) = 1 + X/G(X)$,
where the function $G$$G$ is defined as the solution of the following $\left(1+\rho \right)X=\rho \mathrm{log}\left(\rho \right)\phantom{\rule{2cm}{0ex}}\rho =G\left(X\right)$ What can you say about $X<0$$X<0$? $\phantom{\rule{2cm}{0ex}}$$\hspace{2cm}$ Any guess what is the physical importance of this integral?

### Graphene Cherenkov

Find, prove, or guess the correct analytical expressions for the following
for every real-valued $a$$a$, and especially $|a|<1$$|a|<1$ ${\int }_{0}^{\pi }\frac{{\mathrm{cos}}^{2}\left(\theta \right)\mathrm{sin}\left(\theta \right)}{\mathrm{cos}\left(\theta \right)-a}d\theta$ ${\int }_{0}^{\pi }\frac{{\mathrm{cos}}^{3}\left(\theta \right)}{\mathrm{cos}\left(\theta \right)-a}d\theta$ ${\int }_{0}^{\pi }\frac{{\mathrm{cos}}^{2}\left(\theta \right)}{\mathrm{cos}\left(\theta \right)-a}d\theta$

### Twisted light

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

### Non-perturbative QED

The goal is to find an efficient way to calculate ${P}_{n}\left(t\right)$$P_n(t)$ for large $n$$n$ and an arbitrary t
(${z}_{1},{z}_{2},...,{z}_{n}$$z_1,z_2,...,z_n$ and $x$$x$ sum over all elements in the domains of the corresponding functions ${\mathrm{\Gamma }}_{n}$$\Gamma_n$ and ${G}_{n}$$G_n$, which are defined over domains of size $M$$M$ and are zero everywhere else) ${P}_{0}\left(t\right)=|\sum _{x}^{}{e}^{ixt}{G}_{0}\left(x\right){|}^{2}$ ${P}_{1}\left(t\right)=\sum _{z1}{\mathrm{\Gamma }}_{1}\left({z}_{1}\right)|\sum _{x}^{}{e}^{ixt}{G}_{0}\left(x\right){G}_{1}\left(x-{z}_{1}\right){|}^{2}$ ${P}_{2}\left(t\right)=\sum _{z1}{\mathrm{\Gamma }}_{1}\left({z}_{1}\right)\sum _{{z}_{2}}{\mathrm{\Gamma }}_{2}\left({z}_{2}\right)|\sum _{x}^{}{e}^{ixt}{G}_{0}\left(x\right){G}_{1}\left(x-{z}_{1}\right){G}_{2}\left(x-{z}_{1}-{z}_{2}\right){|}^{2}$ ${P}_{n}\left(t\right)=\sum _{z1}{\mathrm{\Gamma }}_{1}\left({z}_{1}\right)\sum _{{z}_{2}}{\mathrm{\Gamma }}_{2}\left({z}_{2}\right)\dots \sum _{{z}_{n}}{\mathrm{\Gamma }}_{n}\left({z}_{n}\right)|\sum _{x}^{}{e}^{ixt}{G}_{0}\left(x\right){G}_{1}\left(x-{z}_{1}\right){G}_{2}\left(x-{z}_{1}-{z}_{2}\right)\dots {G}_{n}\left(x-{z}_{1}-{z}_{2}-\dots -{z}_{n}\right){|}^{2}$
A naive calculation will have a time complexity O$\left({M}^{n+1}\right)$$(M^{n+1})$, which is inefficient since it grows exponentially in $n$$n$
Find an efficient algorithm for calculating ${P}_{n}\left(t\right)$$P_n(t)$
Hint: can you find an algorithm that has a setup time of complexity O$\left({M}^{3}{n}^{3}\mathrm{log}\left(MN\right)\right)$$(M^3n^3\log(MN))$
with additional O$\left(MnT\right)$$(MnT)$ operations to calculate ${P}_{n}\left(t\right)$$P_n(t)$ for $T$$T$ values of $t$$t$?

### Short times

What is the behavior of $|U\left(t\right)|$$|U(t)|$ at very small $t$$t$?
$U\left(t\right)={\int }_{e}^{\mathrm{\infty }}\frac{{e}^{i\omega t}}{\omega {\mathrm{log}}^{2}\left(\omega \right)}d\omega$