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"... Problem sheet for minicourse ‘Probabilistic Galois Theory’ Q1 In example 2, we considered the interval [1, N] and for each odd prime p ≤ Q = [ √ N] removed all integers n ∈ [1, N] such that ( n p) = −1. It is easy to see that each square survives this process, but do only the squares survive? You ..."

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Problem sheet for minicourse ‘Probabilistic Galois Theory’ Q1 In example 2, we considered the interval [1, N] and for each odd prime p ≤ Q = [ √ N] removed all integers n ∈ [1, N] such that ( n p) = −1. It is easy to see that each square survives this process, but do only the squares survive? You can first discuss this question in the easier setting where for each odd prime p (restriction p ≤ Q dropped) we remove all quadratic non-residues. Q2 Show that once you established Theorem 1 (the large sieve inequality) for one M, you can deduce it for all M, i.e. it was permissible in class without loss of (N + 1)]. generality to assume that M = [ − 1