Treffer: Representation of odd integers as the sum of one prime, two squares of primes and powers of 2: Representation of odd integers as the sum of one prime, two squares of primes and powers of~2

The author addresses the problem of giving a lower bound for the number of powers of two needed to write every large odd integer as \[ N= p_1 + p_2^2 + p_3^2 + 2^{v_1}+ \cdots + 2^{v_k}. \] where \(p_1, p_2, p_3\) are prime numbers. Denoting by \(r_k(N)\) the number of solutions of the equation above, the author proves the following: There exists a constant \(k_0\geq 22000\) and a constant \(N_k\) depending only on \(k\) such that if \(N\geq N_0\), \(k\geq k_0\), then \(r_k(N)>0\). The main technique used on the proof is the circle method.