Śrīnivāsa Rāmānujan

Śrīnivāsa Rāmānujan  born Śrīnivāsa Rāmānujan  Aiyangar,  (22 December 1887 – 26 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.

The number 1729. It is known as Rāmānujan  number. It is the smallest number which can be expressed as the sum of two cubes in two different ways.

1729 = 13+ 123= 93+ 103

Contributions
Rāmānujan  number :The number 1729. It is known as Rāmānujan  number. It is the smallest number which can be expressed as the sum of two cubes in two different ways.

1729 = 13+ 123= 93+ 103

Infinite Series for π : Śrīnivāsa Rāmānujan discovered infinite series for π in1910. The series $$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801}\sum_{k=0}^\infty \frac{(4k\mid)(1103+26390k)}{(k)^4\, 396^{4k}}    $$

Theory of Equations : He derived the formula to solve biquadratic equations.

Asymptotic Formula : He worked on partition of numbers. Using partition function p(n) derived a number of formulae in order to calculate the partition of numbers

$$p(n)\thicksim \frac{1}{4n\sqrt{3}} e^\pi\sqrt{\frac{2n}{3}}, n\rightarrow\infty     $$

Ramanujan’s Congruences :

He discovered the congruences

$$p(5n+4) \equiv 0(mod 5) p(5n+4) \equiv 0(mod 5) p(5n+4) \equiv 0(mod 5)$$