For b ≤ −2 and e ≥ 2, let S e,b : Z → Z ≥0 be the function taking an integer to the sum of the e-powers of the digits of its base b expansion. An integer a is a b-happy number if there exists k ∈ Z + such that S k 2,b (a) = 1. We prove that an integer is −2-happy if and only if it is congruent to 1 modulo 3 and that it is −3-happy if and only if it is odd. Defining a d-sequence to be an arithmetic sequence with constant difference d and setting d = gcd(2, b − 1), we prove that if b ≤ −3 odd or b ∈ {−4, −6, −8, −10}, there exist arbitrarily long finite sequences of d-consecutive b-happy numbers.
This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.
Abstract. This article examines the realizability of small groups of order 2 k , k ≤ 4, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.
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