Fast Bounds on the Distribution of Smooth Numbers

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.

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On the Zeckendorf Representation of Smooth Numbers

Moscow Mathematical Journal ◽

10.17323/1609-4514-2021-21-1-31-42 ◽

2021 ◽

Vol 21 (1) ◽

pp. 31-42

Author(s):

Yann Bugeaud

Keyword(s):

Smooth Numbers

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Smooth Numbers

Equivalents of the Riemann Hypothesis ◽

10.1017/9781108178266.015 ◽

2017 ◽

pp. 332-358

Keyword(s):

Smooth Numbers

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Kloosterman sums over smooth numbers

Journal of Number Theory ◽

10.1016/j.jnt.2017.06.011 ◽

2018 ◽

Vol 182 ◽

pp. 221-235

Author(s):

Zhenzhen Qin ◽

Tianping Zhang

Keyword(s):

Kloosterman Sums ◽

Smooth Numbers

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Evaluation of Penicylinders Used in Disinfectant Testing: Bacterial Attachment and Surface Texture

Journal of AOAC INTERNATIONAL ◽

10.1093/jaoac/70.5.903 ◽

1987 ◽

Vol 70 (5) ◽

pp. 903-906

Author(s):

Eugene C Cole ◽

William A Rutala ◽

Johnny L Carson

Keyword(s):

Stainless Steel ◽

Surface Texture ◽

Environmental Protection Agency ◽

Bacterial Attachment ◽

Dilution Method ◽

Bacterial Inoculation ◽

Protection Agency ◽

Smooth Numbers ◽

Salmonella Choleraesuis ◽

Aoac Method

Abstract Two passible deficiencies in the AOAC use-dilution method for registration of chemical disinfectants by the Environmental Protection Agency are examined: (7) the physical disparities among brands of penicylinders and (2) the variability of bacterial numbers on penicylinders depending upon test strain and penicylinder surface texture. Textual differences of 2 brands of stainless steel penicylinders, one brand of porcelain, and one brand of glass were assessed by scanning electron microscopy. A considerable variation in smoothness of both inner and outer surfaces of stainless steel and porcelain penicylinders was observed. Glass penicylinders were very smooth. Numbers of bacteria attached to a penicylinder were assessed by vortexing the penicylinders 30 s at No. 4 after using the AOAC method of bacterial inoculation and drying 40 min at 37°C. With this methodology, stainless steel carriers retained the 3 AOAC-recommended bacterial test strains differentially: ca 107 for Pseudomonas aeruginosa, 5 x 106 tat Staphylococcus aureus, and 106 for Salmonella choleraesuis; glass retained 10'-107 organisms of all 3 test strains; porcelain retained about that amount of S. aureus but 105-106 P. aeruginosa and 103-104 S. choleraesuis. These data suggest that disinfectants are not similarly challenged with the AOAC-recommended test bacteria and that an alternative method should be considered to ensure comparable numbers of bacteria on penicylinders

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Another note on smooth numbers in short intervals

International Journal of Number Theory ◽

10.1142/s1793042116500196 ◽

2016 ◽

Vol 12 (02) ◽

pp. 323-340 ◽

Cited By ~ 1

Author(s):

Kaisa Matomäki

Keyword(s):

Smooth Numbers ◽

Short Intervals ◽

Prime Factors

In this paper, we prove that, for any positive constants [Formula: see text] and [Formula: see text] and every large enough [Formula: see text], the interval [Formula: see text] contains numbers whose all prime factors are smaller than [Formula: see text].

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