We call a set of natural numbers $\mathcal S$ to be a Sidon Set 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.

We call a set of natural numbers $\mathcal S$ to be a Sidon Set 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.

Gerry Myerson points out OEIS A046881, which gives some really useful results. But, it is through here that I came to know that Euler had a proof that there are infinitely many numbers that can be written as the sum of two fourth powers in two different ways (sourcehttps://math.hawaii.edu/home/talks/resume-fev2011.pdf). But, I couldn't find Euler's proof. Can somebody please refer me to the proof?

The Wikipedia article on Sidon sets mentions

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 want to know how much (if any) progress have been made on any of the two conjectures of Erdős. In particular, how far has the computers checked the second conjecture?

We call a set of natural numbers $\mathcal S$ to be a Sidon Set 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.