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## Tuesday, April 24, 2018

### Some History and Unsolved Problems in Number Theory

Number theory is a very old subject, which is concerned with the set of integers. Number theory started with the concept of integers and simple operations on the integers such as addition, subtraction, etc. Number theory of the greeks is primarily found in the works of Euclid and Plato. Indian mathematicians of antiquity such as Brahmagupta also made significant contributions (one of Brahmagupta's contributions was work on what is now known as Pell's equation). Pierre de Fermat was an important figure in number theory as well, he is responsible for Fermat's theorem as well as Fermat's Last Theorem (a problem which is now solved). Some additional major figures in early number theory were Leonhard Euler, Joseph-Louis Lagrange, and Carl Friedrich Gauss.

Eventually number theory itself started to split into recognizable subbranches, two major ones being algebraic number theory and analytical number theory. Analytical number theory is concerned with the use of real and complex analysis to number theory and can be said to have started with the Dirichlet prime number theorem. Algebraic number theory is concerned with the use of abstract algebra in number theory, and has it origins in reciprocity and cyclotomy.

Despite the significant work done in number theory, there are still plenty of unsolved problems in the field. Many of these problems are easy to understand (although clearly not so easy to solve). Here are a few of them.