# Computational projects

(computo, ergo sum)

## Introduction

Long ago, I decided to find interesting problems, usually requiring extensive verifications and/or enumerations, to occupy the idle CPU time of most workstations and personal computers in my working environment. This page presents links to short summaries of ongoing or finished computational projects I have found interesting and stimulating. Virtually all of them are in one way or another related to the theory of numbers. Most of these pages contain links to selected results of these extensive computations. To save space, download time, and to provide some data integrity checks, these ASCII text files were compressed with the gzip program. These files can be uncompressed using either the gunzip, zcat, or zless programs. Windows users can also uncompress them using, for example, the WinZip, WinRAR, or 7-Zip programs. As far as I am aware, some of the results reported in these pages are (or were for some time) records of computation.

## Project descriptions

The following list contains a brief description of some of my ongoing (circle) or stopped (disk) computational projects.

- The 3x+1 conjecture (alternative link, due to a brain-dead windows server)
- Some generalizations of the 3x+1 conjecture (alternative link, due to a brain-dead windows server)
- Enumeration of polyominoes and other animals
- Determination of the minimum width of admissible prime constellations
- Verification of the Goldbach conjecture
- Numerical experiments related to the prime k-tuple conjecture
- Counts of prime gaps (a by-product of the Goldbach conjecture verification)
- Counts of twin prime gaps
- Counts of Gaussian prime gaps
- Tables with values of the prime counting functions
*pi(x)*and*pi2(x)* - Counts of least primitive roots of prime numbers (Artin's conjecture)
- Search for large fundamental solutions of Pell's equation
- Tables with approximate values of the first zeros on the critical line of some
Dirichlet
*L*-series

**Disclaimer:** Some of the results presented or made available in the pages described above
have not yet been double-checked. They are thus potentially inaccurate. Simple screening tests
were used to identify and reject wrong results. However, these screening tests are not perfect,
and thus wrong results may have been accepted. So far the double-checking computations have
encountered only two wrong results (in the Goldbach conjecture verification project).

Numbers of the form *N·10 ^{M}* will the written either in the form

*N·10^M*or in the form NdM, and numbers of the form

*N·2*will the written either in the form

^{M}*N·2^M*or in the form NbM. For example, 4d18 and 20b58 are the same as

*4·10*and

^{18}*20·2*.

^{58}## List of available software packages related to number theory

Available software packages:

- Fast implementation of the segmented sieve of Eratosthenes

All software is released under the version 2 of the GNU general public license.