3,379 research outputs found
Approximating reals by sums of two rationals
We generalize Dirichlet's diophantine approximation theorem to approximating
any real number by a sum of two rational numbers with denominators . This turns out to
be related to the congruence equation problem with .Comment: 13 pages, improved results and some changes in the proof
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
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