# Number Theory

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## Number Theory Foundations

Number theory is concerned with discovering the amazing and sometimes mysterious relationships between numbers. Review basics of integer and ordered sets, and explore the proof technique of mathematical induction.

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## Divisibility

Elementary students learn about the basic function of division, but divisibility has many more intricate facets not explored in simple mathematics. Begin by reviewing the relationship between numbers and their divisors. Explore LCMs and GCDs and apply Euclid's algorithm. Apply the Fundamental Theorem of Arithmetic to begin investigating prime numbers.

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## Prime Numbers

Prime numbers, as the building blocks of integers, are vastly important in number theory. Explore in the prime number theorem how primes are less common as they become larger and discover different classifications of primes such as Fermat and Mersenne primes. Apply tests to determine the primality of a number through prime factorization.

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## Congruences

Congruences in numbers occur when the difference between two integers is a multiple of some $n$, called the modulus. Use the concept of congruence to explore modular arithmetic and linear congruence equations. Apply modular arithmetic in Wilson's Theorem and Fermat's Little Theorem to discover prime numbers.

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## Arithmetic Functions

Arithmetic functions are any real or complex-valued functions that are defined only on the set of positive integers. These functions are simple, but incredibly useful to number theory. Explore various arithmetic functions and, in particular, the Euler-Phi function, which is used to identify coprime numbers. Compare multiplicative and non-multiplicative functions and apply advanced transforms with the Mobius inversion and the Dirichlet product.

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## The Group of Units

In the Group of Units, explore primitive roots by applying concepts of coprimality and congruence to numbers in modular arithmetic. Learn how to define universal exponents.

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### Universal Exponents

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Quadratics in number theory extend the concept of modular arithmetic to squares. Apply congruences to quadratics, determine if quadratic equations are solvable using quadratic reciprocity, apply residues to help factor large numbers, and explore uses of the Legendre symbol.

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## The Riemann Zeta Function

The Riemann zeta function is applied in number theory to imply results about the distribution of prime numbers. Discover how complex variables are utilized by the Riemann zeta function and how the function can be generalized with the Dirichlet series and Euler products. Lastly, generate random integers using the Riemann zeta function.

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## Sum of Squares

Number theory investigates which integers can be created as a sums of squares using various techniques. Learn about Gaussian integers and constructing integers from sums of four, three, and two squares.

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## Primality Testing

Advanced methods in testing for primality without factorization is very useful for cryptology purposes. Learn to identify numbers that classify as probable primes, how to utilize Carmichael numbers, and how to apply various primality tests.

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## Encryption

One of number theory's most useful applications in this information age is cryptography. Explore the basics of ciphers and the principles underlying public key cryptography.

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