Number theory is a branch of mathematics that deals with the study of integers and their properties. It is a fundamental area of mathematics that has applications in cryptography, computer science, and other areas of science and engineering. Advanced topics in number theory include prime number theory, modular arithmetic, Diophantine equations, and algebraic number theory.
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Key concepts:
The study of prime numbers, their distribution, and properties. This includes the famous prime number theorem, which gives an estimate of the number of prime numbers less than a given number.
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the study of arithmetic operations performed modulo a fixed integer. This concept is important in cryptography, coding theory, and computing.
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The study of equations, typically polynomial equations, with integer solutions. Famous examples include Fermat's Last Theorem, which states that there are no integer solutions to the equation x^n + y^n = z^n for n>2.
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The study of algebraic structures, such as number fields and algebraic integers. This area of number theory involves studying properties of numbers that can be expressed as roots of algebraic equations.
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Lastly, advanced number theory plays a role in diverse fields such as cryptography, computer science, and physics.
