scholarly journals Acceleration of the quadratic sieve method based on the additional search of B-smooth numbers

2018 ◽  
Vol 0 (1) ◽  
pp. 99-106
Author(s):  
Vitalii M. Misko
2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.


2010 ◽  
Vol 7 (7) ◽  
pp. 2193-2198 ◽  
Author(s):  
C. J. Beer ◽  
R. Schiebel ◽  
P. A. Wilson

Abstract. The size-normalised weight (SNW) of planktic foraminifera, a measure of test wall thickness and density, is potentially a valuable palaeo-proxy for marine carbon chemistry. As increasing attention is given to developing this proxy it is important that methods are comparable between studies. Here, we compare SNW data generated using two different methods to account for variability in test size, namely (i) the narrow (50 μm range) sieve fraction method and (ii) the individually measured test size method. Using specimens from the 200–250 μm sieve fraction range collected in multinet samples from the North Atlantic, we find that sieving does not constrain size sufficiently well to isolate changes in weight driven by variations in test wall thickness and density from those driven by size. We estimate that the SNW data produced as part of this study are associated with an uncertainty, or error bar, of about ±11%. Errors associated with the narrow sieve fraction method may be reduced by decreasing the size of the sieve window, by using larger tests and by increasing the number tests employed. In situations where numerous large tests are unavailable, however, substantial errors associated with this sieve method remain unavoidable. In such circumstances the individually measured test size method provides a better means for estimating SNW because, as our results show, this method isolates changes in weight driven by variations in test wall thickness and density from those driven by size.


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