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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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Conjectures on iterated polynomial maps on finite fields
Let $p$ be a prime, and consider the sequence $x_0, x_1, \dots$ of elements of the finite field $\mathbf F_p$ given by $x_0 = 0$ and $x_{i+1} = x_i^2 + 1$ for all $i \ge 0$. This sequence must eventu …