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Roughly speaking, number theory is the study of the integers. Carl Friedrich Gauss is said
to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of
mathematics." Number theorists are interested in topics like the distribution of prime numbers,
the solutions to systems of polynomial equations with integer coefficients, the structure of
symmetry groups of the roots of a polynomial, and the very deep generalizations of these topics.
Many problems in number theory are simple to state but have surprising solutions that draw broadly
from all areas of mathematics, and conjectures in number theory have stimulated major advances
in other fields. Number theory is both beautiful in its abstraction and useful in practice: the foundation
of modern cryptographic systems rely crucially on the difficulty of certain number theoretic problems.