Ramanujan, one of the elegant Mathematician of
India was born in Erode on 22nd December 1887. Erode is a small village (in that
time), 400 Km away from Tamilnadu’s present capital Chennai. His father was a
clerk in Kumbkonam.At the age of five, Srinivasa Ramanujan made his first
appearance in school as a student. It was only a matter of time before it came
to be known that he had extraordinary talent. He showed flashes of brilliance
which were not to be seen in any ordinary kid at that age. He completed his
primary education in a couple of years and then went to Town High School for
further studies. He showed extraordinary liking for mathematics. When he was
yet in school, he mathematically calculated the approximate length of earth's
equator. He very clearly knew the values of the square root of two and the pie
value. At the age of 16, he got scholarship. But his love only for mathematics
cost him the scholarship as he neglected and failed in other subjects. His loss
of scholarship was a great blow to him. He could not afford to study on his
own. He had to find work and leave studies for good. He found a job of an
accounts clerk in the office of the Madras Port Trust. Despite being rejected
two times, his work was recognized by both G. H. Hardy and J. E. Littlewood and
he went to England in 1914. In 1916,he was awarded with a degree of B.Sc.
(later named Ph.D.) by Cambridge University for his work on highly composite
number. In 1916, when he was at his best while working with his colleagues
Hardy & Littlewood, he met with health problems. He was hospitalized in
Cambridge and was diagnosed with T.B. and vitamin deficiency. After two years
struggle, in 1919, he showed some recovery and he decided to return back to
India. However, the improvement was temporary and after his arrival at Bombay,
his health deteriorated again and finally he passed away on 26th April, 1920.His
main contributions in mathematics lie in the field of Analysis, Infinite
Series, Number Theory & Game Theory. His geniusness was that he discovered
his own theorems. Due to his great achievements in the field of Mathematics,
Indian govt. decided to celebrated his birthday 22nd December as Mathematics
Day. ISTE, New Delhi and NBHM, Mumbai have taken initiative to hold
Mathematical competition for students as well as teachers of colleges on the
name of Srinivasa Ramanujan from 2012 to till date so that students and
teachers of India know about the legacy of such great mathematician of India.
I.
Hardy-Ramanujan Number
Once Hardy visited to Putney
where Ramanujan was hospitalized. He visited there in a taxi cab having number
1729. Hardy was very superstitious due to his such nature when he entered into
Ramanujan’s room, he quoted that he had just came in a taxi cab having number
1729 which seemed to him an unlucky number but at that time, he prayed that his
perception may go wrong as he wanted that his friend would get well soon, but
Ramanujan promptly replied that this was a very interesting number as it is the
smallest number which can be expressed as the sum of cubes of two numbers in
two different ways as given below:
1729 =13+123=103+93
Later some theorems were established in theory of elliptic curves which involves this fascinating number.
II. Goldbach’s Conjecture
Goldbach’s conjecture is one of the important illustrations of Ramanujan contribution towards the proof of the conjecture. The statement is every even integer >2 is the sum of two primes, that is, 6 = 3+3.
Ramanujan and his associates had shown that every
large integer could be written as the sum of at most four (Example:
43 = 2+5+17+19).
III. Theory of
Equations
Ramanujan was shown how to solve cubic equations in
1902 and he went on to find his own method to solve the quadratic. He derived
the formula to solve biquadratic equations. The following year, he tried to
provide the formula for solving quintic but he couldn’t as he was not aware of
the fact that quintic could not be solved by radicals.
IV.
Ramanujan-Hardy Asymptotic Formula
Ramanujan’s one of the major work was in the partition
of numbers. By using partition function p(𝑛), he derived a number of formulae in order to calculate the partition
of numbers. In a joint paper with Hardy, Ramanujan gave an asymptotic formulas
for p(𝑛). In fact, a careful analysis of the generating
function for p(𝑛) leads to the Hardy-Ramanujan asymptotic formula
given by
p(n) ~ 1/(4n√3) eπ√(2n/3) , n→∞
In their proof, they discovered a new method called
the ‘circle method’ which made fundamental use of the modular property of the
Dedekind η-function. We see from the Hardy-Ramanujan formula that p(𝑛) has exponential growth. It had the remarkable
property that it appeared to give the correct value of p(𝑛) and this was later proved by Rademacher using
special functions and then Kenono gave the algebraic formula to calculate
partition function for any natural number 𝑛.
Ramanujan’s congruences are some remarkable
congruences for the partition function. He discovered the congruences
(5𝑛+4) ≡0 (𝑚𝑜𝑑 5)
(7𝑛+5) ≡0 (𝑚𝑜𝑑7)
(11𝑛+6) ≡0(𝑚𝑜𝑑 11), ∀𝑛∈𝑁.
In his 1919 paper, he gave proof for the first two
congruences using the following identities using Pochhammer symbol notation.
After the death of Ramanujan, in 1920, the proof of all above congruences
extracted from his unpublished work.
VI. Highly
Composite Numbers
A natural number n is said to
be highly composite number if it has more divisors than any smaller natural
number. If we denote the number of divisors of n by d(n), then we
say 𝑛 є N is called a highly composite if 𝑑 (𝑚) <𝑑(𝑛) ∀𝑚 < 𝑛 where 𝑚, n∊
N. For example, 𝑛 = 36 is highly composite because it has (36) = 9 and
smaller natural numbers have less number of divisors. If
n = 2k2 3k3…pkp (by Fundamental theorem of Arithmetic)
is the prime factorization of a highly composite
number 𝑛then the primes 2,3,..., 𝑝 form a chain of consecutive primes where the sequence
of exponents is decreasing; i.e.𝑘2 ≥ 𝑘3 ≥.....≥ kp and the final exponent 𝑘p
is 1, except for 𝑛 = 4 and 𝑛 = 36.
VII.
Some Other Contributions
Apart from the contributions mentioned above, he
worked in some other areas of mathematics such as hypogeometric series,
Bernoulli numbers, Fermat’s last theorem. He focused mainly on developing the
relationship between partial sums and products of hyper-geometric series. He
independently discovered Bernoulli numbers and using these numbers, he
formulated the value of Euler’s constant up to 15 decimal places. He nearly
verified Fermat’s last theorem which states that no three natural number 𝑥, 𝑦 and 𝑧 satisfy the equation 𝑥n +𝑦n = 𝑧n for any
integer 𝑛 > 2.
