We prove that the number of zeros of random trigonometric polynomials of degree $N$ are normally distributed with mean $(2/\sqrt{3})N$ and variance $cN$, for some constant $c>0$. An analogous result holds in short intervals.

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Let $\phi(t)\in \mathbb Q(t)$ be of degree $d\geq 2$. For a given rational number $x_0$ , define $x_{n+1} = \phi(x_n)$ for each $n\geq 0$. If this sequence is not eventually periodic, and if $\phi$ does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then the numerator of $x_{n+1}-x_n$ has a primitive prime factor for all sufficiently large $n$. The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of $x_{n+1}-x_n$ . Hence the result for each rational function $\phi$ implies (a new proof) that there are infinitely many primes.

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