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


Fibonacci pseudo-primes

Fibonaci number of index n modulo n seems to give ±1 for every prime n (except for n=5)
Can you prove it?
(There are non-primes that still satisfy this condition)
n 1234567891011121314151617
F(n) 11235813213455891442333776109871597
F(n)modn 01110213251011551

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
Can you prove it?
are there non-primes that still satisfy this condition?
n 1234567891011121314151617
L(n) 13471118294776123199322521843136422073571
L(n)modn 01111011431213111

Soliton recursion

The recursion relation below describes two sequences of complex numbers zn and wn
What can you say about these sequences for all n? Try simulating it!

wn+1=Fd(wn,zn) Fg(u,v):=(1g+|v|2)u+gvgvu+1+(1g)|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):=1c+c/c
Note that f has a nice symmetric property: c=f(d)=f(f(c))
Given an arbitrary choice of z0, w0, and c, what is the behavior of the sequences zn and wn?


F(X):=dy((1ey)Xy)2+π2 Prove that F satisfies the following equation for all X>0
1F(X)1+log(1F(X)1)=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+ρ)X=ρlog(ρ)ρ=G(X) What can you say about X<0? 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, and especially |a|<1 0πcos2(θ)sin(θ)cos(θ)adθ 0πcos3(θ)cos(θ)adθ 0πcos2(θ)cos(θ)adθ

Twisted light

Simplify the following integral
02πdϕeikxcos(ϕ)+ikysin(ϕ) More generally, what function is the following integral representing for an integer m?

Non-perturbative QED

The goal is to find an efficient way to calculate Pn(t) for large n and an arbitrary t
(z1,z2,...,zn and x sum over all elements in the domains of the corresponding functions Γn and Gn, which are defined over domains of size M and are zero everywhere else) P0(t)=|xeixtG0(x)|2 P1(t)=z1Γ1(z1)|xeixtG0(x)G1(xz1)|2 P2(t)=z1Γ1(z1)z2Γ2(z2)|xeixtG0(x)G1(xz1)G2(xz1z2)|2 Pn(t)=z1Γ1(z1)z2Γ2(z2)…znΓn(zn)|xeixtG0(x)G1(xz1)G2(xz1z2)…Gn(xz1z2…zn)|2
A naive calculation will have a time complexity O(Mn+1), which is inefficient since it grows exponentially in n
Find an efficient algorithm for calculating Pn(t)
Hint: can you find an algorithm that has a setup time of complexity O(M3n3log(MN))
with additional O(MnT) operations to calculate Pn(t) for T values of t?

Short times

What is the behavior of |U(t)| at very small t?