1
$\begingroup$

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?

Improve this question
$\endgroup$
7
  • 2
    $\begingroup$ I think there has been no progress toward deciding whether $x^5+y^5=z^5+w^5$ has any nontrivial solutions (other than perhaps raising the lower bounds for existence by computer search). $\endgroup$
    – Gerry Myerson
    Commented Dec 23, 2023 at 22:24
  • 2
    $\begingroup$ The number of solutions to $n= x^5+y^5=z^5+w^5$ in integral $x,y,z,w$ can be seen to be at most $\ll_{\epsilon} n^{\epsilon}$ from the divisor bound. The Bombieri-Lang conjecture implies the number of solutions are in fact bounded. See the comments in the post mathoverflow.net/questions/232574/… for more discussion. $\endgroup$
    – Mark Lewko
    Commented Dec 23, 2023 at 22:33
  • 1
    $\begingroup$ According to Guy, Unsolved Problems in Number Theory, people have searched up to $x^5+y^5=10^{25}$ for nontrivial solutions to $x^5+y^5=z^5+w^5$. Perhaps the search has gone farther in the years since that book was published. $\endgroup$
    – Gerry Myerson
    Commented Dec 24, 2023 at 16:35
  • 1
    $\begingroup$ At oeis.org/A046881 it says, "Randy Ekl discovered that a number that can be written in two ways as a sum of two fifth powers exceeds $4.01\times10^{30}$. - R. J. Mathar, Sep 07 2017 "According to the Mathworld links below, a(5), if it exists, exceeds $1.02\times10^{26}$. The page at the SquaresOfCubes link below says Stuart Gascoigne did an exhaustive search and found in Sep 2002 that no a(5) solution less than $3.26\times10^{32}$ exists. - Jon E. Schoenfield, Dec 15 2008 "$a(5) > 10^{33}$. - Julien Courties, Nov 02 2020" $\endgroup$
    – Gerry Myerson
    Commented Dec 24, 2023 at 16:45
  • 2
    $\begingroup$ I think the relevant paper of Euler is: arxiv.org/pdf/math/0505629.pdf $\endgroup$
    – Mark Lewko
    Commented Dec 25, 2023 at 0:05

0

Reset to default

You must log in to answer this question.