Ramanujan’s Elementary Method in Partition Congruences
نویسنده
چکیده
Page 182 in Ramanujan’s lost notebook corresponds to page 5 of an otherwise lost manuscript of Ramanujan closely related to his paper providing elementary proofs of his partition congruences p(5n + 4) ≡ 0 (mod 5) and p(7n+ 5) ≡ 0 (mod 7). The claims on page 182 are proved and discussed, and further results depending on Ramanujan’s ideas are established.
منابع مشابه
Elementary proofs of congruences for the cubic and overcubic partition functions
In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan’s cubic continued fraction. Chan proved that ∑ n≥0 a(3n+ 2)q = 3 ∏ i≥1 (1− q3n)3(1− q) (1− qn)4(1− q2n)4 which clearly implies that, for all n ≥ 0, a(3n+ 2) ≡ 0 (mod 3). In the same year, Byungchan Kim introduced the overcubic partition function a(n). Using modular forms, Kim proved that ∑ n≥0 a(3n +...
متن کاملCongruences for Andrews’ Spt-function
Congruences are found modulo powers of 5, 7 and 13 for Andrews’ smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo ` for all primes ` ≥ 5 which were conjectured earlier by the author. We extend Ono’s method to handle the powers of...
متن کاملCongruences for Andrews ’ Spt - Function modulo Powers of 5 , 7 and 13
Abstract. Congruences are found modulo powers of 5, 7 and 13 for Andrews’ smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo for all primes ≥ 5 which were conjectured earlier by the author. We extend Ono’s method to handle the pow...
متن کاملPartition congruences by involutions
We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences. Introduction The theory of partitions is a beautiful subject introduced by Euler over 250 years ago and is still under intense development [2]. Arguably, a turning point in its history was the invention of the “constructive partition theory” symbolized by F...
متن کاملPowers of Euler’s Product and Related Identities
Ramanujan’s partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)∞ satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r , the coefficients in the expansions of (q; q)∞ satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011