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Abstract. This page is by no means exhaustive, but here are some sites, softwares, organizations and individual efforts that have been inspiring or particularly beneficial to me in my quest. Most of the descriptions are based on material written at the original site.
Software, web references, research, databases, educational materials and notable efforts to encourage mathematical discovery.
The website arXiv.org, in it's own words, is "an e-print service in the fields of physics, mathematics, non-linear science, computer science, and quantitative biology. The contents of arXiv conform to Cornell University academic standards. arXiv is owned, operated and funded by Cornell University, a private not-for-profit educational institution. arXiv is also partially funded by the National Science Foundation."
The The Beal Conjecture, by Andrew Beal who resides in Dallas, Texas where he owns and operates Beal Bank and Beal Aerospace Technologies, Inc. Mr Beal has proffered $100,000 of his own money into a trust held by the American Mathematical Society as a prize to anyone who can resolve his mathematical conjecture, to wit, suppose Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.
The Clay Math Institute has the stated primary objectives
The freeware program Fractint is fractal generator created for IBM PCs and compatible computers to run under DOS and ported to Linux. I originally became acquainted with this program in the early days when the folks behind it were known as the Stone Soup Group and the original Compuserve was their exclusive home on the net. To my mind, this program was "the" original fractal generating software and The Stone Soup Group was one of the early champions of the freeware concept.
The award-winning MacTutor History of Mathematics Archive is an integrated collection of over 1000 biographies and historical articles of a mathematical nature, alongside interactive birthplace maps and an index of famous curves, most with interactive Java applets.
Wolfram Research, the makers of Mathematica Software, maintain the website MathWorld which bills itself as the web's most extensive mathematics resource. They may be right about that. I hope to make original contributions to this site in the near future.
Rob Sacks is an amateur mathematician who published this little gem of a web site, NumberSpiral.com about the same time I was posting about my personal discovery of the "The Square Spine Spiral", which of course turned out to be a variation of Ulam's Spiral, which is really just a single specific example of more general quadratic behavior.
Now while I, as far as I still know to this day, can continue to claim to be the first to discover and produce 3D versions of the spiral that elegantly include all of the integers in cohesive, numerical order, Rob produced essentially the same thing, just flatter. He also did a far superior job of detailing the math behind it. In the subsequent years, I have been witness to scores of amateurs proclaiming to have found a pattern to the distribution of prime numbers and almost inevitably, what these folks are seeing is very well known to all number theorists as the inevitable consequences of the Sieve of Eratosthenes and/or Quadratic Reciprocity.
Generally speaking, the pretty pictures and patterns are of little practical use to mathematicians. However, I am a firm believer that they are invaluable as an educational tool and that from amongst those who discover, become fascinated by and pursue this beauty on their own time, of their own volition, one shall find persons who have all the makings and motivations to become great mathematicians.
The GPL licensed PARI/GP computer algebra system is designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, transcendental functions and more.
The online resource Plouffe's Inverter is web accessible database of more than 215,000,000 mathematical constants like π, e, Catalan or the Euler-Mascheroni constant, γ.
The Persistence of Vision Raytracer is a high-quality, totally free tool for creating stunning three-dimensional graphics. It is available in official versions for Windows, Mac OS/Mac OS X and i86 Linux. The source code is available for those wanting to do their own ports. This is the program I used to make the 3D pictures of the Square Spine Spiral.
The PrimeForm discussion group is for the primality-testing program PrimeFormGW (PFGW). The list's purpose is to discuss anything prime-related -- theory, algorithms, methods, and other computer programs are suitable topics.
The website Sloane's Encyclopedia of Integer Sequences is a resource for information about a particular number sequence. You can ask the database if it can recognize your favorite sequence, if you have one. If you have an interesting sequence, Sloane's may have additional information about it, or it may not be entered, in which case you can stake your claim by submitting it and describing how it is generated. The data base is approaching 100,000 entered sequences as of the time of this writing.
TeX to HTML (TtH) is a program that translates TEX, the predominant mark-up language for expressing mathematics, into HTML, the language of world-wide-web browsers. It produces more compact, faster viewing, web documents than other converters, because it really translates the equations, instead of converting them to images. TtH is free for non-commercial applications. (use the -u switch to produce output readable by FireFox)
A great resource, W3Schools.com is for anyone getting started on any aspect of web development. This site includes basic tutorials and serves as a good quick reference while you're working. The depth of coverage of each subject is enough to give a solid, basic understanding, but what is really impressive is the breadth of topics covered--HTML, XML, browser scripting languages, server scripting languages and more. If you write web pages, or are learning how, check it out.
The program WinShell is a free multilingual integrated development environment (IDE) for LaTeX and TeX. The program includes a text editor, syntax highlighting, project management, spell checking, a table wizard, BibTeX support, different toolbars and user configuration options. It is not a LaTeX system, it's the software you need to be able to use a LaTeX system.
The website XahLee.org has over 1200 HTML pages of original content and is visited by 4,000-6,000 unique visitors per day. The site is the effort of one individual, Xah Lee as you may have guessed, and features a comprehensive visual dictionary of plane curves along with animations, visualizations and a veritable variety pack of other interesting information mathematical and otherwise.
The Prime Pages website is "the web resource" for comprehensive and up to date information about prime numbers. The site was originally compiled by Samuel Yates as place to keep an up to date list of the largest known prime numbers, a function the site has continuously performed to this day. It is perhaps best known for it's list of the 5,000 largest known prime numbers. Under the tutelage and loving care of Professor Chris Caldwell, who continued Yate's work over a decade(?) ago, it has become a leading web resource on all things having to do with prime numbers.
The CafePress.com website allows one to upload print designs for clothing and other products, which are then printed on demand and shipped to the customer who just purchased it on-line through CafePress' secure web site. CafePress is where the Imathination shops were created and now reside.
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