I think Elkies' construction of numbers $a,b,c,d$, such that $a^4 + b^4 + c^4 = d^4$, disproving Euler's sum of powers conjecture, is quite impressive. Another nice construction from number theory is Mahler's construction of numbers $n$ such that $r_{3,3}(n) > cn^{1/12}$, where $r_{3,3}(n)$ denotes the number of representations of $n$ as sum of three non-negative cubes, which disproved Hypothesis K of Hardy and Littlewood.