منابع مشابه
Partition 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 کاملCongruences of the Partition Function
Ramanujan also conjectured that congruences (1) exist for the cases A = 5 , 7 , or 11 . This conjecture was proved by Watson [17] for the cases of powers of 5 and 7 and Atkin [3] for the cases of powers of 11. Since then, the problem of finding more examples of such congruences has attracted a great deal of attention. However, Ramanujan-type congruences appear to be very sparse. Prior to the la...
متن کاملThe Andrews–Sellers family of partition congruences
In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers' conjecture for all powers of 5. In addition, we discuss why the Andrews-Sel...
متن کاملRamanujan’s Elementary Method in Partition Congruences
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.
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1923
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1923-03753-1