A probabilistic test for equality a = bc for given n-bit integers a, b, c is designed within complexity n(log logn) exp{O(log∗ n)}.

Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to ...

We show that the set of composite positive integers n ≤ x satisfying the congruence (2n−1 n−1 ) ≡ 1 (mod n) is of cardinality at most x exp ( −(1/ √ 2 + o(1)) √ log x log log x ) as x →∞.

Let 9 (x) denote the pseudoprime counting function. With L(x) = exp{log x log log log x/log log *}> we prove 9(x) < x ■ L(x)~'/2 for large x, an improvement on the 1956 work of Erdös. We conjecture that 9(x) = xL(x)"1+o(1).

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent ∈ H (0, 1). While the periodicity ensures that the ultimate long time behavior is diffusive, the generalized Sinai potential considered here leads to a strong logarithmic...

Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class Gp(N) of random graphs, the corresponding ßooding time is investigated. The ßooding time TN(p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the ßooding time TN(p) converges in distribution to the two-fold convolution Λ(2∗) of the Gumbel distribut...

A minimization (resp., maximization) problem is called fixed parameter (r, t)approximable for two functionsr, t if there exists an algorithm that given an integer k and a problem instance I with optimum value opt, finds either a feasible solution of value at most r(k) · k (resp., at least k/r(k)) or a certificate that k < opt (resp., k > opt), in time t(k) · |I|O(1). A problem is called fixed p...