On Diophantine equations with Fibonacci and Lucas sequences associated to Hardy-Ramanujan problem

In this paper, we deal with Diophantine equations $N={F_k}^3+{F_{\mathrm{\ell }}}^3={F_m}^3+{F_n}^3$ and $M={L_k}^3+{L_{\mathrm{\ell }}}^3={L_m}^3+{L_n}^3$. In other words, we discover the Fibonacci and Lucas numbers that are also Hardy-Ramanujan numbers.