Contributions of Srinivasa Ramanujan to the Number Theory
“An equation for me has no meaning, unless it expresses a thought of God.”
L. Littlewood described Ramanujan with the words `Every positive integer is one of his personal friends.’ The expression amply describes the heights in the realm of numbers to which Ramanujan rose. Here are few contributions made by Ramanujan to the number theory.
Magic Squares
As a young student, Srinivasa Ramanujan was interested in magic squares. The simplest magic square problem is to fill up the cells in a square with 3 rows and 3 columns with the numbers 1; 2, 3, …….. , 9 such that each row sum = each column sum = each diagonal sum. One can find the following solution:
4  9  2 
3  5  7 
8  1  6 
In this magic square, each row sum = each column sum = each diagonal sum = 15. This magic square requires an understanding of how 15 can be written as a sum of 3 nonrepeated numbers. One can think of magic squares of types 4 × 4, 5 × 5……..: Ramanujan gave the following formula for a general magic square of type 3 ×3:
C+Q 
A+P  B+R 
A+R  B+Q  C+P 
B+P  C+R  A+Q 
Where A, B, C (respectively P, Q, R) are integers in arithmetic progression. The following formula was also provided by him.
2Q+R  2P+2R  P+Q 
2P  P+Q+R  2Q+2R 
P+Q+2R  2Q  2P+R 
Theory of Partitions
The magic squares form the nucleus of the theory of partitions developed by Srinivasa Ramanujan. His fascination for magic squares led him in his later life to work on this theory. Let us consider the partitions of a natural number. Let p(n) denote the partition function n, defined as the number of ways of expressing n as a sum of natural numbers ≤ n.
For example, 1 has the partition 1; 2 have the partitions 2, 1+1; 3 has the partitions 3, 2+1, 1+1+1, and so on. As n increases, p(n) becomes larger and larger. For example, 6 has the partitions 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1.
The following table provides the values of p(n) for n = 1, 2,……, 20 .
Table 1:
n  1  2  3  4  5  6  7  8  9  10  
p(n)  1  2  3  5  7  11  15  22  30  42  
n  11  12  13  14  15  16  17  18  19  20  
p(n)  56  77  101  135  176  231  297  385  490  627 
Denoting the number of partitions of n with parts ≤ m by pm(n) we have the recurrence relation pm(n) = pm1(n) + pm( n –m ) (1 < m ≤ n). About the partitions of a natural number, G. H. Hardy and E. M. Wright remarked in [5], “in spite of the simplicity of the definition of p(n), not very much is known about its arithmetic properties”. Under this back ground, it is worthwhile to consider the contributions of Ramanujan.
Three papers by Ramanujan on the theory of partitions were published in the years 1919, 1920 and 1921.
Ramanujan’s Congruences
When m is 0 or a natural number, Ramanujan obtained the following congruences: p(5m + 4) ≡ 0 (mod 5), p(7m + 5) ≡ 0 (mod 7), p(11m + 6) ≡ 0 (mod 11). Regarding the contribution of Ramanujan to the theory of partitions, G. H. Hardy and E. M. Wright wrote “The simplest arithmetic properties were found by Ramanujan. Examining Mac Mahon’s table of p(n); he was first led to conjecture and then to prove, three striking arithmetic properties associated with the moduli 5, 7 and 11 “. To illustrate the results of Ramanujan, the values of p(n) for n ≡ 4 (mod 5), 5 (mod 7), 6 (mod 11) are provided in tables 2, 3, 4, respectively.
Table 2:
n

4

9

14

19

24

29

34

39


p(n)  5  30  135  490  1575  4565  12310  31185 
Table 3:
n  5  12  19  26  33  40 
p(n)  7  77  490  2436  10143  37338 
Table 4:
n  6  17  28  39 
p(n)  11  297  3718  31185 
n a natural number is said to be composite if it has a divisor different from 1 and itself. Ramanujan raised an interesting question: If n is a composite number, what makes it a highly composite one? For this purpose, he considered the number of distinct positive divisors of n denoted by d(n).
Definition: Highly composite number (Srinivasa Ramanujan [10])
A natural number n is a highly composite number if d(m) < d(n) for all m < n.
When one considers the primes and composite numbers in Z, 1 is a unit element and 2 is a prime. However, both of them become highly composite numbers as per the definition of Ramanujan. The first few highly composite numbers and the number of their distinct positive divisors are enumerated in the following table.
Table 5:
n  1  2  4  6  12  24  36  48  60  120  180 
d(n)  1  2  3  4  6  8  9  10  12  16  18 
n  240  360  720  840  1260  1680  2520  5040  7560  10080  
d(n)  20  24  30  32  36  40  48  60  64  72 
One of the highly composite numbers calculated by Ramanujan is 6 7 4 6 3 2 8 3 8 8 8 0 0.
This number has 13 digits and its prime factorization is 2^{6}. 3^{4}. 5^{2}. 7^{2}. 11. 13. 17. 19. 23.
Theorem: Form of a highly composite number (Ramanujan [10])
If n = 2^{a}^{1}3^{a}^{2}5^{a}^{3}……P^{a}^{p} is highly composite number, then a1 ≥ a2 ≥ a3 ≥…….≥ ap and ap = 1 except for n = 4 and 36.
It is to be noted that 4 = 2^{2 }and 36 = 2^{2}. 3^{2}
^{ }
Theorem: Successive highly composite numbers are asymptotically equivalent.
Srinivasa Ramanujan published a paper on highly composite numbers in 1915. He was awarded B. A. Degree by research by Cambridge University in 1916 for his dissertation titled `Highly composite numbers’ and his advisors were G. H. Hardy and J. E. Littlewood.
Formulae for π:
For the transcendental number π Ramanujan has given several formulae. Some of them are listed below:
π= 24 tan^{1 }1/8 + 8 tan^{1} 1/57+4 tan^{1}1/239
π ≈ 63/15. 17+15√15/7+15√15
Diophantine Equations
Algebraic equations requiring solutions in integers are called Diophantine equations. They are named after the mathematician of Alexandria, Diophantus. For Diophantine equations, one may refer to L. J. Mordell. Let us briefly see the contributions of Srinivasa Ramanujan to Diophantine equations.
Ramanujan’s Number
The number 1729 has acquired a special status in mathematics. It is referred to as Ramanujan number. There is a famous anecdote about this number. Srinivasa Ramanujan made a statement to G. H. Hardy that 1729 is the smallest number that can be expressed as a sum of two cubes in two different ways. We have the two expressions 1729 = 9^{3} + 10^{3} and 1729 = 1^{3 }+ 12^{3}. Thus 1729 is the smallest integral solution of the equation A^{3 }+ B^{3 }= C^{3}+ D^{3}.
Ramanujan gave the general solution of this equation as
( a + x^{2}y)^{3} + ( xβ + y )^{3 }= ( Xa + y )^{3 }+ ( β + X^{2}Y)^{3} where a^{2} + aβ + β^{2} = 3XY^{2}.
The Diophantine equation: X^{3}+Y^{3}+Z^{3} = W^{3}
Ramanujan found out the following parametric solution to the Diophantine equation X^{3}+Y^{3}+Z^{3} = W^{3}
X = 3a^{2 }+ 5ab – 5b^{2}
Y = 4a^{2} – 4ab + 6b^{2}
Z = 5a^{2} – 5ab – 3b^{2}
W = 6a^{2} – 4ab +4b^{2}
^{ }
Symmetric Equation
Srinivasa Ramanujan considered the Diophantine equation x^{y} = y^{z}. One can observe the symmetry of this equation in x and y. Ramanujan proved that this equation has only one integral equation i.e., x = 4 and y = 2 and there are an infinite number of rational solutions.
Ramanujan’s Equation
The Diophantine equation X^{2} + 7 = 2^{n} is called Ramanujan’s equation. He gave the solutions 1^{2} + 7 = 2^{3}, 3^{2}+ 7 = 2^{4}, 5^{2} + 7 = 2^{5}, 11^{2}+7= 2^{7}, 181^{2} + 7 = 2^{15} and conjectured that the above equation has no other integral solution. This equation remained unsolved until 1948.Working in the quadratic number field Q (√ – 7). T. Nagell proved Ramanujan’s conjecture in 1948. He published the English version of this paper in 1961.
Ramanujan’s equation has a bearing on triangular numbers and Mersenne numbers. A triangular number has the form m (m + 1) / 2. Numbers of the form 2^{k} – 1 are known as Mersenne numbers.
The Utility of Ramanujan’s Equation
Ramanujan’s equation finds applications in coding theory. One may refer to the papers by H. S. Shapiro and L. D. Slotnick, R. Alter and E. L. Cohen. Several results pertaining to the generalization of Ramanujan’s equation have been described by E. L. Cohen and the author.
References:
[1] R. Alter, On a Diophantine equation related to perfect codes, Mathematics of
Computation, 25 (115), 621624, 1971.
[2] B. C. Berndt, Y. S. Choi and S. Y. Kang, The problems submitted by Ramanujan to
Indian Mathematical Society, Contemporary Mathematics, 236, 215256, 1999.
[3] E. L. Cohen, Sur l’ equation Diophantine x2 + 11 = 3k; C. R. Acad. Sci. Paris Ser. A,
275, 57, 1972.
[4] E. L. Cohen, On the RamanujanNagell equation and its generalizations in Number
Theory: Proceedings of the First Conference of the Canadian Number Theory
Association, 8192, 1990.
[5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford
Clarendon Press, Fourth Edition, Oxford, 1960.
[6] R. Kanigel, The man who knew infinity, A life of the genius Ramanujan, Washington
Square Press, Washington, 1992.
[7] L. J. Mordell, Diophantine Equations, Academic Press, London, 1969.
[8] T. Nagell, Losning till oppgave nr 2, 1943, Norsk Mathematisk Tidsskrift, 30, 6264,
1948.
[9] T. Nagell, The Diophantine equation x2 + 7 = 2n; Arkiv for Mathematik, 4, 185187,1961.
[10] S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14, 347409, 1915.
[11] S. Ramanujan, Question 464, The Journal of Indian Mathematical Society, 5, 120, 1919.
[12] S. Ramanujan, Collected papers of Ramanujan, Cambridge University Press, Cambridge,
1927.
[13] S. Ramanujan, Collected papers, Cambridge Chelsea Publishing Co., New York, 1962.
[14] S. Ramanujan, Collected Papers of Srinivasa Ramanujan, (Ed. G. H. Hardy, P. V. S.
Aiyar, and B. M. Wilson). Amer. Math. Soc., Providence, Rhode Island, 2000.
[15] A. M. S. Ramasamy, Ramanujan’s equation, Journal of Ramanujan Mathematical Society,
7 (2), 133153, 1992.
[16] H. S. Shapiro and L. D. Slotnick, On the Mathematical theory of error correcting codes,
IBM Journal of Research and Development, 3 (1), 2534, 1959.
Acknowledgments
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