Number Theory and Diophantine Equations in India

Aryabhata (476-550 CE) is considered to be the father of Indian number theory. He made significant contributions to the field, including the development of a system of decimal notation and the discovery of the quadratic formula.

Brahmagupta (598-668 CE) is known for his work on Diophantine equations. He developed a method for solving linear Diophantine equations and also studied quadratic Diophantine equations.

The Hardy-Littlewood conjecture states that every positive integer can be expressed as the sum of three prime numbers. It was first proposed by G. H. Hardy and J. E. Littlewood in 1923 and remains unproven.

Srinivasa Ramanujan (1887-1920) is known for his work on modular forms. He discovered many remarkable properties of modular forms and made significant contributions to the field of number theory.

Pell's equation is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^2 - y^2 = n$. It was first studied by the Indian mathematician Brahmagupta in the 7th century CE.

Brahmagupta (598-668 CE) is known for his work on the Fibonacci sequence. He discovered the recurrence relation for the Fibonacci sequence and also studied its properties.

Fermat's Last Theorem is a theorem that states that for any positive integer $n > 2$, there exist no three positive integers $x$, $y$, and $z$ such that $x^n + y^n = z^n$. It was first proposed by Pierre de Fermat in the 17th century CE and remained unproven for over 350 years. It was finally proven by Andrew Wiles in 1994.

Srinivasa Ramanujan (1887-1920) is known for his work on the Basel problem. He discovered a remarkable formula for the sum of the reciprocals of the squares of the natural numbers, which helped to solve the Basel problem.

Catalan's conjecture is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n - y^n = z^2$. It was first proposed by Eugène Charles Catalan in the 19th century CE and remains unproven.

Srinivasa Ramanujan (1887-1920) is known for his work on the Riemann hypothesis. He discovered many remarkable properties of the Riemann zeta function and made significant contributions to the study of the Riemann hypothesis.

The Bombieri-Vinogradov theorem is a theorem that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^2 + y^2 = z^n$. It was first proven by Enrico Bombieri and A. I. Vinogradov in the 20th century CE.

Srinivasa Ramanujan (1887-1920) is known for his work on the Waring's problem. He discovered many remarkable properties of the Waring's problem and made significant contributions to the study of the problem.

Catalan's conjecture is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n + y^n = z^m$. It was first proposed by Eugène Charles Catalan in the 19th century CE and remains unproven.

Srinivasa Ramanujan (1887-1920) is known for his work on the Goldbach's conjecture. He discovered many remarkable properties of the Goldbach's conjecture and made significant contributions to the study of the conjecture.

Catalan's conjecture is an equation that states that for any positive integer $n$, there exist infinitely many pairs of integers $x$ and $y$ such that $x^n - y^n = z^m$. It was first proposed by Eugène Charles Catalan in the 19th century CE and remains unproven.