Algorithmic Number Theory is a branch of mathematics that focuses on developing algorithms to solve problems related to integers and their properties. It combines elements of number theory, which studies the properties of numbers, with computational techniques to efficiently perform calculations and solve equations involving whole numbers. This field has practical applications in areas such as cryptography, where secure communication relies on the difficulty of certain number-theoretic problems. Key topics include prime numbers, greatest common divisors, and modular arithmetic, all of which are essential for creating algorithms that can handle large numbers and complex calculations.