In the twentieth century, progress was finally made on this question, which had been left untouched for over 100 years. Building on work of Tomio Kubota, S. J. Patterson and Roger Heath-Brown in 1978 disproved Kummer conjecture and proved a modified form of Kummer conjecture. In fact they showed that there was equidistribution of the θp. This work involved automorphic forms for the metaplectic group, and Vaughan's lemma in analytic number theory. In 2000 further refinements were attained by Heath-Brown.
