SCHWEITZER 2007 Problem 7

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$ , such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm.

(translated by Miklós Maróti)