computations, on ENIAC. The calculations were programmed and coded by Hedvig Selberg but her work was only acknowledged at the end of the paper, similarly as with Mary Tsingou on the Fermi–Pasta–Ulam–Tsingou problem (formerly the Fermi–Pasta–Ulam problem). 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. Cassels' conjecture A second conjecture on Kummer sums was made by J. W. S. Cassels, again building on previous ideas of Tomio Kubota. This was a product formula in terms of elliptic functions with complex multiplication by the Eisenstein integers. The conjecture was proved in 1978 by Charles Matthews. Patterson's conjecture In 1978 Patterson conjectured that θp was equidistributed with error term asymptotically of order instead of quadratic as with Gauss sums which could explain the initial bias observed by Kummer. Next year his subsequent work with Heath-Brown disproving Kummer's conjecture showed that in fact it was equidistributed, but whether the order of the asymptotic was correct remained unknown. More than 20 years later, Heath-Brown closed on the problem, giving a new sieve method, and conjectured that it could be improved to obtain the predicted order. In 2021 the problem was demonstrated conditionally on the generalized Riemann hypothesis by Alexander Dunn and Maksym Radziwill, who also showed that the sieve of Heath Brown could not be improved as expected. References Cyclotomic fields