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Ramanujan congruences

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1: 27.14 Unrestricted Partitions
§27.14(v) Divisibility Properties
Lehmer (1947) conjectures that τ ( n ) is never 0 and verifies this for all n < 21 49286 39999 by studying various congruences satisfied by τ ( n ) , for example: …
2: Bibliography R
  • S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.
  • 3: 27.20 Methods of Computation: Other Number-Theoretic Functions
    A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    4: Bibliography
  • A. Adelberg (1996) Congruences of p -adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
  • C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter 16 of Ramanujan’s second notebook: Theta-functions and q -series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.
  • G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) Ramanujan Revisited. Academic Press Inc., Boston, MA.
  • G. E. Andrews and D. Foata (1980) Congruences for the q -secant numbers. European J. Combin. 1 (4), pp. 283–287.
  • G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.