DOI: 10.5281/zenodo.5914768
title: Smooth numbers in large prime gaps

description: <strong>Overview</strong> This dataset contains numbers from 25 up to 1 quadrillion (10<sup>15</sup>) that are smooth relative to the gap to the preceding prime. More precisely, we list all numbers <em>n</em> so that <em>r</em> + <em>p<sup>a</sup></em> ≤ <em>n</em> where <em>r</em> is the largest prime smaller than <em>n</em> - 1, and <em>p<sup>a</sup></em> is the largest prime-power divisor of <em>n</em>. The dataset is the result of a 10 day computation using 15 cores on an Intel Xeon system, running code hosted at GitHub (see "Related identifiers"). The GitHub code checks additional conditions when <em>r</em> is <em>n</em> - 2 and <em>n </em>- 1 is a power of 2, but it is easy and quick to check that when (up to 10<sup>15</sup>) <em>n</em> = 2<sup><em>k</em></sup> + 1, the second largest prime <em>r<sub>2</sub></em> satisfies <em>r<sub>2</sub></em> + <em>p<sup>a</sup></em> &gt; n. Thus, this additional check makes no difference in the output. Our motivations for computing this data are described in our paper <em>On invariable generation of alternating groups by elements of prime and prime power order</em> (arXiv:2201.12371). Any number <em>n</em> in the range which is not of the given form has the associated alternating group <em>A<sub>n</sub></em> generated by any element of order <em>r</em> together with any element having a certain cycle structure (and of order <em>p<sup>a</sup></em>). <strong>Description / specification</strong> The data is stored as compressed text-based input to a computer algebra system, specifically in gzipped GAP format. The file out-<em>k</em>.g.gz holds numbers in the range from (<em>k </em>- 1)⋅10<sup>12</sup> to <em>k</em>⋅10<sup>12</sup>. The first line of each file sets the variable invgen_oversmooth_range to be the range (thus, [(<em>k </em>- 1)⋅10<sup>12</sup> .. <em>k</em>⋅10<sup>12</sup>]). The subsequent lines set invgen_oversmooth to a list of pairs of numbers [<em>n</em>, <em>p<sup>a</sup></em>], where <em>n</em> is a smooth number as described above, and <em>p<sup>a</sup></em> is the largest prime-power of <em>n</em>. The largest prime preceding <em>n</em> - 1 is given in a GAP comment. Thus, the first few lines of out-0.g.gz (when uncompressed) appear as <pre><code>invgen_oversmooth_range:=[25..1000000000000]; invgen_oversmooth := [ [ 30, 5 ], # bp 23 [ 60, 5 ], # bp 53 [ 126, 9 ], # bp 113 [ 210, 7 ], # bp 199 [ 252, 9 ], # bp 241 [ 308, 11 ], # bp 293 [ 330, 11 ], # bp 317 [ 420, 7 ], # bp 409 ...</code></pre> where [25 .. 1000000000000] is the range considered, and for example "[ 30, 5 ], # bp 23" represents that 23 is the largest prime preceding 30 - 1, 5 is the largest prime-power divisor of 30, and 23 + 5 ≤ 30. We created the data in GAP files for ease of inputting into a GAP program in our own use of the data. It is easy to convert the GAP files to another format via standard technique such as regular expression-based search and replace. For example, on macOS or Linux, the following command will convert the list in out-0.g.gz to a CSV file, which it will display on the terminal. <pre><code class="language-bash">zcat out_quadrillion/out-0.g.gz | sed -En 's/ \[ ([0-9]+), ([0-9]+) \], # bp ([0-9]+)/\1,\2,\3/gp' | less</code></pre>
description: Work of the first author was partially supported by NSF Grant DMS-1901595 and Simons Foundation Fellowship 609771. Work of the second author was supported in part by NSF Grant DMS 1518389. Work of the third author is supported in part by the Slovenian Research Agency research program P1-0285 and research projects J1-9108, N1-0160, J1-2451, J3-3003.