In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Ramanujan's congruences are the congruences for the
partition function ''p''(''n'') discovered by
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar
(22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
:
:
In plain words, the first congruence means that if a number is 4 more than a multiple of 5, i.e. it is in the sequence
: 4, 9, 14, 19, 24, 29, . . .
then the number of its partitions is a multiple of 5.
Later, other congruences of this type were discovered, for numbers and for
Tau-functions.
Background
In his 1919 paper, he proved the first two congruences using the following identities (using
q-Pochhammer symbol
In the mathematical field of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer ...
notation):
:
He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these".
After Ramanujan died in 1920,
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on ''p''(''n'') (Ramanujan, 1921). The proof in this manuscript employs the
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalize ...
.
In 1944,
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...
defined the
rank function for a partition and conjectured the existence of a
"crank" function for partitions that would provide a
combinatorial proof In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof:
* A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in ...
of Ramanujan's congruences modulo 11. Forty years later,
George Andrews and
Frank Garvan
Francis G. Garvan (born March 9, 1955) is an Australian-born mathematician who specializes in number theory and combinatorics. He holds the position Professor of Mathematics at the University of Florida. He received his Ph.D. from Pennsylvania Sta ...
found such a function, and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7 and 11.
In the 1960s,
A. O. L. Atkin of the
University of Illinois at Chicago
The University of Illinois Chicago (UIC) is a public research university in Chicago, Illinois, United States. Its campus is in the Near West Side community area, adjacent to the Chicago Loop. The second campus established under the Universi ...
discovered additional congruences for small prime moduli. For example:
:
Extending the results of A. Atkin,
Ken Ono
Ken Ono (born March 20, 1968) is an American mathematician with fields of study in number theory. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia.
Early life and education
Ono ...
in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give
:
Later
Ken Ono
Ken Ono (born March 20, 1968) is an American mathematician with fields of study in number theory. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia.
Early life and education
Ono ...
conjectured that the elusive crank also satisfies exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg in his 2005 paper ''Partition Congruences and the Andrews–Garvan–Dyson Crank'', linked below. This paper won the first
Proceedings of the National Academy of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Scie ...
Paper of the Year prize.
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 by considering the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
of the following
function in the
l-adic topology:
:
It is seen to have dimension 0 only in the cases where ''ℓ'' = 5, 7 or 11 and since the partition function can be written as a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of these functions this can be considered a formalization and proof of Ramanujan's observation.
In 2001, R.L. Weaver gave an effective algorithm for finding congruences of the partition function, and tabulated 76,065 congruences. This was extended in 2012 by F. Johansson to 22,474,608,014 congruences,
one large example being
:
References
*
*
*
External links
*{{cite journal , last1=Mahlburg , first1=K., url=http://math.mit.edu/~mahlburg/preprints/mahlburg-CrankCong.pdf, title= Partition Congruences and the Andrews–Garvan–Dyson Crank, journal=
Proceedings of the National Academy of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Scie ...
, volume=102, issue=43, pages= 15373–76, year=2005, doi= 10.1073/pnas.0506702102, pmid= 16217020, pmc=1266116, bibcode=2005PNAS..10215373M, doi-access=free
Dyson's rank, crank and adjoint A list of references.
Theorems in number theory
Srinivasa Ramanujan
Equivalence (mathematics)