- 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
Interested in joining my research group at the Technion?
- If you solved 3 puzzles or more, I strongly encourage you to apply to my group
Fibonaci number of index modulo seems to give for every prime (except for )
Can you prove it?
(There are non-primes that still satisfy this condition)
Lucas numbers are much nicer than their Fibonacci relatives:
here it seems that
, for every prime
, with no exceptions
Can you prove it?
are there non-primes that still satisfy this condition?
The recursion relation below describes two sequences of complex numbers and
What can you say about these sequences for all ? Try simulating it!
The parameter is an arbitrary complex number (imaginary part of zero typically gives degenerate dynamics),
and , with
Note that has a nice symmetric property:
Given an arbitrary choice of , , and , what is the behavior of the sequences and ?
Prove that satisfies the following equation for all
Or, in other words, prove that ,
where the function is defined as the solution of the following
What can you say about ? 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 , and especially
Simplify the following integral
More generally, what function is the following integral representing for an integer ?
The goal is to find an efficient way to calculate for large and an arbitrary t
( and sum over all elements in the domains of the corresponding functions and ,
which are defined over domains of size and are zero everywhere else)
A naive calculation will have a time complexity O, which is inefficient since it grows exponentially in
Find an efficient algorithm for calculating
Hint: can you find an algorithm that has a setup time of complexity O
with additional O operations to calculate for values of ?
What is the behavior of at very small ?