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In this paper, we compute the size of the exceptional set in a generalized Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$ with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0 \leq i, j < g$, there are infinitely many integers $n$ which satisfy $f(n) = Ap + Bq$ for some primes $p \equiv i, q \equiv j \pmod{g}$ under a mild condition., Comment: 18 pages

Gross and Zagier conjectured that if the analytic rank of a rational elliptic curve is 1, then the order of the rational torsion subgroup of the elliptic curve divides the product of Tamagawa number, Manin constant, and the square root of the order of Tate--Shafarevich group over an imaginary quadratic field. In this paper, we show that this conjecture is true., Comment: 22 pages; major revision: proved the whole conjecture without any other conjecture, deleted some routine computations (will be given upon request)