We improve the “sieve” part of the number field sieve used in factoring integer and computing discrete logarithm. The runtime of our method is shorter than that of existing methods. Under some reasonable assumptions, we prove that it is less than two-thirds of the running time of the algorithm used before asymptotically with probability greater than 0.6.

We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59 81]. We here extend this to k ≥ 2 and generalize it to all l-tuples of Hawkins primes.

! R e s e a r c h S t a t i o n , C a n a d a D e p a r t m e n t o f A g r i c u l t u r e , Lethbridge, A l b e r t a . samples has been removed by the construction and use of the apparatus described herein. The apparatus consists of an iron plate, 33 X 33 X 0.6 cm, fitted with three upright iron rods, each 4 4 c m long (Fig. 1). Two aluminum rods clamped across the upright rods near the top ...

The distribution of solutes in the various cells of sugar beet (Beta vulgaris L.) source leaves, petioles, and sink leaves was studied in tissue prepared by freeze-substitution. The differences in degree of cryoprotection indicated that sieve elements and companion cells of the source leaf, petiole, and sink leaf contain a high concentration of solute. The osmotic pressure of various types of c...