Consecutive residues or non-residues in the Gaussian integers
نویسندگان
چکیده
منابع مشابه
On the constant in Burgess’ bound for the number of consecutive residues or non-residues
We give an explicit version of a result due to D. Burgess. Let χ be a non-principal Dirichlet character modulo a prime p. We show that the maximum number of consecutive integers for which χ takes on a particular value is less than { πe √ 6 3 + o(1) } p1/4 log p, where the o(1) term is given explicitly.
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Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest possible number of consecutive quadratic non-residues modulo p is always less than p. (One can find a brief discussion of this problem in R. K. Guy’s book [5]). Schur also pointed out that the greatest number of consecutive q...
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In [2] it was shown that A(&, 4) = co for fe^ 1048909 and it was conjectured that A(&, 4) = =° for all k. In this paper we establish this conjecture with the following Theorem. A(&, 4) = ». Proof. It suffices to prove the theorem for values of k which are prime. The proof makes use of the following proposition which is a special case of a result of Kummer [l] (see also [3]). Proposition. Let k ...
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By some extremely simple arguments, we point out the following: (i) If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n. (ii) Let OK be the ring of algebraic integers in a quadratic field K = Q( √ d) with d ∈ {−1,−2,−3,−7,−11}. Then, for any irreducible π ∈ OK and positive integer k no...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1969
ISSN: 0022-314X
DOI: 10.1016/0022-314x(69)90008-0