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 |
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)modn |
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
Can you prove it?
are there 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 |
L(n) |
1 | 3 | 4 | 7 | 11 | 18 | 29 | 47 | 76 | 123 | 199 | 322 | 521 | 843 | 1364 | 2207 | 3571 |
L(n)modn |
0 | 1 | 1 | −1 | 1 | 0 | 1 | −1 | 4 | 3 | 1 | −2 | 1 | 3 | −1 | −1 | 1 |
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!
zn+1=Fc(zn,wn)
wn+1=Fd(wn,zn)
Fg(u,v):=(1−g+|v|2)u+gvg―vu+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/―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?
Entropy
F(X):=∫∞−∞dy((1−e−y)X−y)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
∫π0cos2(θ)sin(θ)cos(θ)−adθ
∫π0cos3(θ)cos(θ)−adθ
∫π0cos2(θ)cos(θ)−adθ
Twisted light
Simplify the following integral
∫2π0dϕeikxcos(ϕ)+ikysin(ϕ)
More generally, what function is the following integral representing for an integer m?
∫2π0dϕe−imϕeikxcos(ϕ)+ikysin(ϕ)
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(x−z1)∣∣∣2
P2(t)=∑z1Γ1(z1)∑z2Γ2(z2)∣∣∣∑xeixtG0(x)G1(x−z1)G2(x−z1−z2)∣∣∣2
Pn(t)=∑z1Γ1(z1)∑z2Γ2(z2)
∑znΓn(zn)∣∣∣∑xeixtG0(x)G1(x−z1)G2(x−z1−z2)
Gn(x−z1−z2−
−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?
U(t)=∫∞eeiωtωlog2(ω)dω