There exist no Ramanujan congruences mod 6912

1975 ◽  
Vol 17 (2) ◽  
pp. 148-153
Author(s):  
A. A. Panchishkin
Keyword(s):  
Ramanujan Congruences

Ramanujan Congruences For p-k(n) Modulo Powers Of 17

1991 ◽  
Vol 43 (3) ◽  
pp. 506-525 ◽  
Author(s):  
Kim Hughes
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Ramanujan Congruences ◽  
Image Position

For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.

scholarly journals Ramanujan congruences for p-k (mod 11')

1983 ◽  
Vol 24 (2) ◽  
pp. 107-123 ◽  
Author(s):  
Basil Gordon
Keyword(s):  
Generating Function ◽  
Euler Product ◽  
Ramanujan Congruences ◽  
Image Position

Denote bythe Euler product, and bythe partition generating function. More generally, if k is any integer, putso that p(n) = p−1(n). In [3], Atkin proved the following theorem.

scholarly journals “RAMANUJAN’S CONGRUENCES AND DYSON’S CRANK”

2014 ◽  
Vol 2 (3) ◽  
pp. 10-32
Author(s):  
Sabuj Das
Keyword(s):  
Theoretical Description ◽  
Undergraduate Mathematics ◽  
Cambridge University ◽  
Congruence Modulo ◽  
Freeman Dyson ◽  
Combinatorial Result ◽  
Ramanujan Congruences ◽  
The Many ◽  
College Library

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11.  Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for P (n), the number of partitions of n indeed, Ramanujan’s formulas lay unread until 1976 when Dyson found In the Trainty College Library of Cambridge University among papers from the estate of the late G.N.Watson. In 1986, F.Garvan wrote his Pennsylvania state Ph.D. Thesis Precisely on the formulas of Ramanujan relative to the crank. In view of this theoretical description, the story of the crank is a long romantic tale and the crank functions are intimately connected to all partitions congruences. In 2005, Mahlburg stated that the crank functions themselves obey Ramanujan type congruences.

Ramanujan Congruences for p-k(n)

1968 ◽  
Vol 20 ◽  
pp. 67-78 ◽  
Author(s):  
A. O. L. Atkin
Keyword(s):  
Partition Function ◽  
Ramanujan Congruences ◽  
Image Position ◽  
Congruence Properties

Let12Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

scholarly journals RAMANUJAN CONGRUENCES FOR A CLASS OF ETA QUOTIENTS

2010 ◽  
Vol 06 (04) ◽  
pp. 835-847 ◽  
Author(s):  
JONAH SINICK
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Generating Functions ◽  
Complete Characterization ◽  
Ramanujan Congruences

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 ( mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences cN(n) satisfy, where cN(n) is defined by [Formula: see text]. This last result answers a question of H.-C. Chan.

scholarly journals Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

2011 ◽  
Vol 63 (6) ◽  
pp. 1284-1306 ◽  
Author(s):  
Michael Dewar
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Half Plane ◽  
Modular Forms ◽  
Generating Functions ◽  
Open Questions ◽  
Ramanujan Congruences ◽  
Upper Half Plane

AbstractRamanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.

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