Let r, s, n be integers satisfying 0 ≤ r < s < n, s ≥ n, α > 1/4, and gcd(r, s) = 1. Lenstra showed that the number of integer divisors of n equivalent to r (mod s) is upper bounded by O((α − 1/4)). We re-examine this problem; showing how to explicitly construct all such divisors and incidentally improve this bound to O((α−1/4)−3/2).

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 1020 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 1018.

Wedescribe a polynomial-size conic quadratic reformulation for amachine-job assignment problemwith separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation. © 2009 Elsevier B.V. All rights reserved.

= 1 or −1 according as j is or is not a quadratic residue mod p. A multivariable generalization of Theorem 1.1 follows. Theorem 1.1 is a special case of Theorem 1.2 with x3 = · · · = xp = 0. Theorem 1.2. Let p be an odd prime and p = (−1)(p−1)/2p. Then there exist integer polynomials R(x1, x2, . . . , xp) and S(x1, x2, . . . , xp) such that 4 · det(circ(x1, x2, . . . , xp)) x1 + x2 + · · ·+ xp ...

So far the only method we have to solve the propositional satisfiability problem is to use truth tables, which takes exponential time in the formula size in the worst case. In this lecture we show that for Horn formulas and 2-CNF formulas satisfiability can be decided in polynomial time, whereas for 3-CNF formulas satisfiability is as hard as the general case. We also show that if we replace di...