Let p(n) denote the number of partitions of n . Ramanujan's partition congruences are p(5n + 4) , p(7n + 5) and p(11n + 6) = mod 5, 7, and 11, respectively. These have been proved in number of ways. Atkin and Swinnerton-Dyer proved the congruences and some more relations about partition İn the case of mod5 and 7 in terms of rank, Garvan proved them in three cases in terms of crank. In this study, we give an another proof of their results in the case of mod5 by using the theory of modular forms. Although our method is more tedious and complicated, it shows us how Modular forms of integral weight on a certain subgroups of SL_2(Z) play role in partition theory. Our method could be applied to the case mod7, but not mod11 since the components of (bak) are not known explicitly.
EKİN, A. Bülent (1997) "The Rank and the Crank Modulo 5," Turkish Journal of Mathematics: Vol. 21: No. 2, Article 5. Available at: https://journals.tubitak.gov.tr/math/vol21/iss2/5