On Shank's algorithm for modular square roots.
We show that the large sieve is optimal for almost all exponential sums.
We prove that there are only finitely many positive integers such that there is some integer such that is 1 or a prime for all , thus solving a problem of Byeon and Stark.
We introduce the concept of asymptotic stability for a set of complex functions analytic around the origin, implicitly contained in an earlier paper of the first mentioned author (“Finite group actions and asymptotic expansion of ", Combinatorica 17 (1997), 523 – 554). As a consequence of our main result we find that the collection of entire functions with the set of all real polynomials satisfying Hayman’s condition is asymptotically stable. This answers a question raised in loc. cit.
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