The Number of Relations in the Quadratic Sieve Algorithm

The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relations is then computed by a known formula. Hence practical hints for the optimal choice of the parameter values can be derived. We further compare theoretical estimates for the total number of smooth integers in an interval with countings in practice. AMS Subject Classification (1991): 11A51, 11Y05 CR Subject Classification (1991): F.2.1 Keywords & Phrases: Factorization, Multiple Polynomial Quadratic Sieve, Vector supercomputer, Cluster of work stations 1. Introduction We assume that the reader is familiar with the multiple polynomial quadratic sieve algorithm [Bre89, Pom85, PST88, Sil87, RLW89]. We consider the single large prime variation of the algorithm and write MPQS for short. If we ..

Pneumothorax following ERCP: Report of four cases and review of the literature

We report four patients with pneumothorax as a complication of ERCP with sphincterotomy. With conservative treatment all patients recovered. Previously, 16 comparable cases have been reported in the literature. The main risk factor for this rare complication seems (pre-cut) sphincterotomy. Pneumothorax is usually right-sided or bilateral and accompanied by pneumomediastinum, pneumoretroperitoneum and subcutaneous emphysema. The prognosis seems favourable with a non-surgical approach including intravenous antibiotics, fasting and when indicated chest tube drainage