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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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Hypergeometric identities
Treating both sides as coefficients of $x^k$ in a power series with variable $x$, the left-hand side turns into a power-series product, one factor representing $\sqrt{1-x}$. We end up with a special c …