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Let $p_i$ be the $i$th prime, and let $f(N)$ be the largest cardinality of a subset of \[ \left\{\prod_{i=1}^N p_i^{a_i}:0\le a_i<N,\ a_i\in\mathbb{Z}\right\} \] containing no two distinct legs of an integer right triangle. We prove \[ f(N) \ll N^N (\log\log N)^{-1/1296}. \]
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, w...
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