DOI: 10.5281/zenodo.5595336
title: 1585 Algebraic numbers to be found for Smale's Problem #7

description: Smale's Problem #7 Solutions Set for n=1..65 (and 2 other force laws)<br> -----------------------------------------------------------------------<br> <br> Introduction:<br> ------------<br> <br> Smale's Problem #7 asks for an arrangement of n-points on a S2 sphere, such that the pairwise sum of all the reciprocals of the logarithm of the distance of any 2 points is minimized. This is equivalent to the J. J. Thomson Problem in physics, or minimizing the potential energy for all points. See https://en.wikipedia.org/wiki/Smale%27s_problems<br> <br> Smale asks for the global minimal soluion, not just a local minimum solution which can be infinite in number.<br> <br> In the datasets given, not only was the log force law answers found, but also the Coloumb attraction (1/r) and the inverse square law (1/r^2) force laws solutions for the points N=1..65 also found as very large floating point numbers or the spherical code for that point.<br> <br> Thus a set of 1585 datafiles consisting of very large spherical codes up to 60,013 digits in length are provided in hopes that an algorithm such as the QLLL or the Lenstra–Lenstra–Lovász lattice basis reduction algorithm can successfully recover the algebraic number associated with its spherical code for these unknown cases.<br> <br> An example of recovering an algebraic number from a large floating point number:<br> <br> n = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912...<br> <br> Running the GP-Pari algdep() command on this number yields<br> <br> ? algdep(n,2)<br> %1 = x^2 - 2<br> <br> This is a successful recovery which went from a floating point number n, to the polynomial mononomial equation x^2-2 of which n is a root. Thus the floating point numbers or spherical codes are roots of a polynomial equation in a single variable.<br> <br> The polynomial is thus considered the algebraic number completely giving n, the floating point number root.<br> <br> Motive of the dataset:<br> ---------------------<br> <br> By providing these 50,014 digit floats, it is hoped that the algebraic number can be successfully obtained, thus finding the minimal global solution to Smale's Problem #7. The math paper cited shows that these floating point numbers are the approximations to the global minimal solutions. 49 algebraic numbers have been found, it is hoped to complete these 109 algebraic numbers for the whole range of 1-65 points.<br> <br> This is a challenge to the currently existing mathematics software as the degree of the algebraic numbers &gt; 420. New faster algorithms are needed to recover them.<br> <br> -----------------<br> <br> Description:<br> -----------<br> <br> The database contains the following file set:<br> <br> <br> 1. Master index csv file<br> <br> Name: parameter_sizes.csv<br> <br> This master list gives the size of N points, total count of parameters, the name of each parameter, its digit size, and polynomial degree check.<br> <br> There are from 2 to 40 parameters necessary for a given N size. Most of the parameters are 50,014 digits in size.<br> <br> <br> 2. Parameter files - 1585 ASCII text files<br> <br> Name of file: law + count + parameter id<br> <br> These 1,585 files consisting of 3 basic sets based upon the potential field or force law affecting the points:<br> <br> 1r for 1/r force law<br> 2r for 1/r^2 force law<br> log for log force law<br> <br> Each parameter is named after the law, the count of points and a-z,A-Z suffixed to delineate the parameter itself:<br> <br> law + point count + unique id<br> (1r, 2r, log) + [1..65] + [a-z,A-Z]<br> <br> Example:<br> <br> log law, 13 points, 2nd parameter --&gt; log13b.para<br> <br> ---------------------------<br> - parameter file format -<br> ---------------------------<br> <br> The ASCII text format of log13b.para: (same format for ALL parameter files)<br> This is a very high precision decimal float or spherical code<br> <br> ------------------------------------------------------------------------<br> 0.50004268876.... (usually 50014 digits) ....2100512063619728807073247<br> ------------------------------------------------------------------------<br> <br> There are 109 distance point sets in 3 force law categories, for a total of 1,585 parameter files.<br> <br> <br> 3. Resolved parameters - 10 ASCII text files<br> <br> The following 10 parameters have been successfully resolved to their algebraic number, but are kept in place with the other unresolved parameters for this particular count of N and the force law.<br> <br> 1r53b.para.resolved<br> 2r19a.para.resolved<br> 2r19b.para.resolved<br> 2r53b.para.resolved<br> log24a.para.resolved<br> log24b.para.resolved<br> log24c.para.resolved<br> log24e.para.resolved<br> log24i.para.resolved<br> log53b.para.resolved<br> <br> It is hoped that the other 1,585 parameters in the set can be successfully resolved.<br> <br> Citation:<br> --------<br> <br> Please see https://arxiv.org/abs/2008.04880<br> <br> Some spherical codes in S2 and their algebraic numbers<br> Randall L Rathbun, Wesley JM Ridgway<br> <br> Software:<br> --------<br> <br> The GP-Pari program was used to find some of the algebraic numbers, specifically the algdep() command. See https://pari.math.u-bordeaux.fr/ for this excellent number theory software<br> <br> Correspondence Author:<br> ---------------------<br> <br> Randall L Rathbun (randallrathbun@protonmail.com)<br>
description: Three extra files are included: README, metadata and parameter_sizes.csv files. The parameter_sizes.csv files indexes the parameters and gives their digit size for a N-point under one of 3 potentials (force law).