The orbits of 4583 main belt asteroids are integrated orbits for 57 years and searched for asteroid-asteroid encounters from which it may be possible to determine the masses of 23 of the largest asteroids (diameter>=200km) and 11 smaller asteroids. The search is conducted using a scattering formula which serves as a useful filter for identifying encounters that can lead to a mass determination. A total of 460 such encounters were found. The ten most useful of these encounters are examined in detail. The results show that, to make a reliable mass determination, the mean distance of the perturbed asteroid must be known to within a few times 10^-8^AU. An observing program targeting the asteroids listed here would have a substantial impact on our knowledge of asteroid masses and densities.