Tiling number problem and $\boldsymbol{\theta}$-congruent number problem

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.