**As promised, here are a wealth of links to
some of the very best mathematical web sites from around the globe, and all at
the click of a mouse.**

**The online encyclopaedia of integer sequences**

If you are interested in integer sequences and their properties, this is the definitive site and easily the best resource available on the internet. It is maintained and run by Neil Sloane, and I personally have written, commented on or corrected over 250 sequences. To test it out, simply copy a sequence such as 25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116 into the search box provided and press ENTER.

This is a superb website which offers a complete index of all the most famous curves in the history of mathematics. Once you have accessed the site, simply select the name of the curve you wish to see and it will appear on your screen along with its associated Cartesian and parametric representation. For those of you with web browsers that support JAVA applets, this site also allows interactive experimenting with the curves. Simply alter the parameters and see the effect it has on the behaviour of the curve.

**Topics in the History of Mathematics**

Select this
site to visit an excellent work on the history and development of
Numbers and Number Theory in mathematics. Links are provided from the times of
the ancient Babylonians through to the present day, culminating in Andrew
Wiles’ famous proof of Fermat’s Last Theorem.

For those readers that are interested in the mathematicians
themselves, there is a very nice biographical
site that is entirely devoted to this topic. There is also an A-Z of mathematicians that have appeared on postage
stamps.

**Eric Weisstein’s World of Mathematics**

The mathworld site is hailed by Internet users worldwide to be the web’s most extensive mathematical resource, and is hosted by Wolfram Research, the makers of Mathematica. The alphabetical A-Z index a particularly useful reference.

Recently, Wolfram Research have launched Wolfram Alpha which they describe as a “computational knowledge engine”. This is arguably the best resource for students on the internet and is well worth the effort of learning how to use it properly. For example, type integrate 1/(1-x^3) into the box provided, press ENTER and it instantly evaluates the integral. Moreover, selecting the “show steps” option gives all the working out that you need for a manual evaluation. But it doesn’t just stop there. For example, try entering the sequence 1,4,9,16,25,… (with the ellipsis marks) and see what happens. Or you might like to ask it “What is the mass of the moon?”. The possibilities appear endless.

**Numbers that are simultaneously square and triangular**

A problem that has fascinated recreational mathematicians
throughout the ages has been to find a relationship between those integers that
are both square and triangular. The mathworld site gives an excellent
treatment of this problem and includes a general formula, a recurrence
relation for generating the sequence, a product formula and a generating function.
The first few elements of the sequence constitute sequence A001110
in the OEIS**.**

For the interested
reader, Sloane’s A036353 gives the first Square
Pentagonal Numbers, A036354 gives the first Square
Heptagonal Numbers, A036428 gives the first Square
Octagonal Numbers and A036411 gives
the first Square Nonagonal Numbers.

This site prides itself on collecting mathematically related
material currently available on the internet, and provides a gateway for
accessing all the mathematical discussion groups. Whilst on this site, you may
care to spend “a few moments” solving *the
problem of the week*.

Describing itself as “A Gateway to Modern Mathematics”, this site includes material on a selection of topics as diverse as number theory and differential equations. Under each topic heading there are links to related internet sites and references to standard texts on the subject matter.

**Assistance with solving mathematical
problems**

If you are really stuck you may be able to obtain guidance from the mathnerds, purple maths or Dr. Math. However, do not under estimate the usefulness of Wolfram Alpha (see above).

**Mathematical Associations**

Here are a few links to mathematical associations and societies from around the world. The American Mathematical Society, the Mathematical Association of America, the Society for Industrial and Applied Mathematics, the Mathematical Sciences Research Institute, the Institute of Mathematics and its Applications (UK), the British Society for the History of Mathematics, the European Mathematical Society, the London Mathematical Society, the Mathematical Association (UK) and last (but not least) the Royal Statistical Society (UK).

**Euclid’s Elements**

In order to know where we are going, it is essential to know
where we have come from. This web site discusses the famous works of Euclid of
Alexandria, circa 300 BC, which are the foundation stones of modern geometry.

**Calculus**

Traditionally, calculus is a branch of mathematics that students have always found difficult. However, there is plenty of help available on the internet and the links that follow should provide a platform to allow you to resolve any particular difficulties that you might come across. A particularly useful site is at calculus.net, and this provides links to many other good sites. If its graphics that you want, then Douglas Arnold’s Graphics for the Calculus Classroom offers a wide variety of options including a link to its companion site Graphics for Complex Analysis. But once again, do not under estimate the usefulness of Wolfram Alpha (see above).

**The search for prime numbers**

Ever since Euclid
of Alexandria, circa 300 BC, proved that there are infinitely many prime
numbers, mankind has been obsessed with searching for larger and larger
primes. On ^{rd}
August, 2008^{th}
(known) Mersenne
Prime and is given by *M*_{43112609
}= 2^{43112609}-

The Prime Pages (probably the best source of reference on the Internet) offer prime number research, records and resources.

The **G**reat **I**nternet **M**ersenne **P**rime **S**earch, GIMPS. Take a look at the people
and the machinery behind our quest to find larger and larger primes, and
register your chance to share in the $50000 award for
finding the first prime number with 100 million or more digits. Landon
Curt Noll also maintains a complete record of all the currently known
Mersenne primes.

Goldbach’s
conjecture is one of the simplest
results to understand, yet has the distinction of being one of the most sought
after proofs in number theory today. Can you prove that every even integer
greater than 2 can be expressed as the sum of two prime numbers? To generate publicity for the novel *Uncle Petros and Goldbach's
Conjecture* by Apostolos Doxiadis, British publisher Tony Faber
offered a $1,000,000 prize if a proof was submitted before April 2002. The
prize was not claimed.There are many other famous, seemingly
trivial unsolved problems in number theory, and the subject is so widely
studied perhaps for no other reason than this.

**Fibonacci Numbers**

Next to prime numbers, Fibonacci numbers
are arguably amongst the most popular in number theory today. Named after Leonardo of Pisa
(1170-1240), the sequence of Fibonacci numbers (Sloane’s
A000045) was originally generated as the solution to a problem regarding pairs of breeding
rabbits. Subsequently, it has become one of the most studied sequences in
the history of mathematics and has been found to be related to what appear to
be extremely diverse topics. Examples include the golden
ratio, ancient Greek architecture, music, flowers and
the diagonals
of a pentagon. The relationship between each of the above topics was the
subject of the 168^{th}
Christmas Lecture (in 1997), and only the second ever on mathematics.

There are many identities
for the Fibonacci numbers, some of which connect them with the
transcendental numbers *e*, π and
the imaginary *i*. Other related
sequences include the Lucas
sequence, Sloanes
A000204. And finally, if you have found these sites sufficiently
interesting, why not consider subscribing to the Fibonacci Quarterly, a journal that serves as
a focal point for interest in Fibonacci

numbers and related questions, especially with respect to new results.

**Fun with Mathematical Miscellany**

For a large collection of recreational topics, check out The World of Numbers.

Here is an award winning proof of Pythagoras’ Theorem (by a Java applet).

Use the Mandelbrot Explorer to investigate the delightful world of the Mandelbrot set, and try not to leave before you check out the images available for download.

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