We call a set of natural numbers $\mathcal S$ to be a [*Sidon Set*](https://en.wikipedia.org/wiki/Sidon_sequence) if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.

Erdős conjectured that there exists a nonconstant integer-coefficient polynomial whose values at the natural numbers form a Sidon sequence. Specifically, he asked if the set of fifth powers is a Sidon set.

Ruzsa came close to this by showing that there is a real number $c$ with $0<c<1$ such that the range of the function
$$f(x)=x^5+\left\lfloor cx^4\right\rfloor$$
is a Sidon sequence.

I mainly have two (related) questions :-

 1. How much (if any) progress have been made on Ruzsa'a result? Are there any other almost-polynomials that have been found which may be able to do the job?

 2. How far has the computers checked the second conjecture? Related to this is the question : what algorithms do we know of (better that brute force) to check whether a given set is a Sidon set or not? Checking all the cases (ie., brute forcing) won't be a good idea since $100^5=10^{10}$ is already quite large.