**Ramanujan's Congruences for the Partition Function modulo 5, 7, 11**

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**Ayşegül Yavuz**

Mathematics, MSc. Thesis, 2018

**Thesis Jury**

Assoc. Prof. Dr. Kağan Kurşungöz (Thesis Advisor),

Asst. Prof. Dr Ayesha Asloob Qureshi,

Asst. Prof. Dr. Zafeirakis Zafeirakopoulos

**Date & Time:** 12th December, 2018 – 13.30 PM

**Place: **Fens 2008

**Keywords : **Jacobi's triple product identity, integer partition, Ramanujan's congruences, Winquist's identity, q-series

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**Abstract**

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In 1919, Ramanujan introduced three congruences satisfied by the partition function p(n), namely p(5n+4) ≡ 0 mod 5, p(7n+5) ≡ 0 mod 7 and p(11n+6) ≡ 0 mod 11. In this thesis, our aim is to present different proofs of each of these congruences. For the congruence p(5n+4) ≡ 0 mod 5, we present three types of proofs from elementary to non-elementary. We observe that the elementary proofs of the congruences p(5n+4) and p(7n+5) are analogues with minor variations. The second proof of the congruences p(7n+5) can be regarded as non-elementary proof. Even though their non-elementary proofs are similar to each other, the proof of the congruence in modulo 7 involves considerably more computations on identities, inevitably, than the proof of the congruence in modulo 5. We further present three worth-stressing proofs for the congruence p(11n+6) ≡ 0 mod 11; The first is proved by Winquist using a representation of as a double series and a two parameter identity is utilized for this double sum. Then Hirschhorn proves this congruence using a four parameter generalization of Winquist's identity and modifies the representation of . Lastly, owing to Ramanujan, B. Berndt, et al., prove the congruence p(11n+6) ≡ 0 mod 11 directly using a new representation for . We complete the thesis by presenting a more direct and a uniform proof, given by Hirschhorn in 1994, that can be applicable to all three congruences. This proof is partially based on linear algebra, which makes it reasonably different from Winquist's and Hirschhorn's earlier proofs.