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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
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Enumerating ways to decompose an integer into the sum of two squares
Another point of view, which might be easier from an algorithmic point of view (in my opinion), is to look at the decomposition $n = a^2+b^2$ using complex numbers $n = (a+bi)(a-bi)$. Suppose you know …