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Let $F_{k}(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is $F_5(N)=(1-o(1))N$? More generally, is $F_{2k+1}(N)=(1-o(1))N$?

Conjectured by Erdős, Sós, and Sárkzözy [ErSaSo95], who proved this for $F_3(N)$. Erdős [Er35] earlier proved that $F_4(N)=o(N)$. Erdős also asks about $F(N)$, the size of the largest such set such that the product of no odd number of $a\in A$ is a square. Ruzsa proved that $\lim F(N)/N <1$. The value of $\lim F(N)/N$ is unknown, but it is $>1/2$.

Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$.

Erdős [Er51] proved that, for infinitely many $k$,
\[ n_{k+1}-n_k \gg \frac{\log n_k}{\sqrt{\log\log n_k}}.\]
Richards [Ri82] improved this to
\[\limsup_{k\to \infty} \frac{n_{k+1}-n_k}{\log n_k} \geq 1/4.\]
The constant $1/4$ here has been improved, most lately to $0.868\cdots$ by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard [DEKKM22].
The best known upper bound is due to Bambah and Chowla [BaCh47], who proved that
\[n_{k+1}-n_k \ll n_k^{1/4}.\]