# Sums

Tip: SOME OF THE LINKS FROM THIS WEBSITE OPEN IN A NEW WINDOW. IF AT FIRST YOU DON'T SUCCEED, OR YOU GET A "POP-UP BLOCKED" MESSAGE, PRESS AND HOLD CTRL AND CLICK AGAIN.

### Pythagoras theorem

You probably vaguely remember Pythagoras theorem - the one that says that, for any triangle, the sum of the square on the hypotenuse is equal to the sum of the squares on the other two sides. But can you prove it?

Here is a delightful interactive proof of Pythagoras theorem, and a brilliant example of the Internet as an educational resource.

### Mandelbrot set

"The Mandelbrot Set is a mathematically generated picture of unfathomable complexity and startling beauty" according to my son Alistair's website. Don't take his word for it. Have a look at his Mandelbrot generator. The set takes about thirty seconds to appear. You can then explore it following his instructions. It is very beautiful, but is it Art?

### Prime numbers and factorization

Even the simplest calculators handle addition, subtraction, multiplication and division. Multiplying whole numbers together can be done in a fraction of a second. For example, 3*367 = 1,101. However, breaking a large whole number into its prime factors is another matter altogether. What, for example, are the prime factors of 1,102? Or 1,103?

In fact, 1,102 = 2*19*29, but it would take some time to discover this by trial and error. And 1,103 is a prime number, having no factors (apart from 1 and itself). Finding the factors of very large numbers by trial and error is simply not practicable, even using the fastest computers.

One of the most remarkable programs I have found on the Web is is Dario Alpern's Factorization program. This program uses serious Number Theory to factorize unimaginably large numbers incredibly quickly. Don't be put off by the scary title. You simply type your number in the box, and press return.

Here are a few interesting whole numbers to try:

2,047   2,048   20,112,011   2,147,483,647  147,573,952,589,676,412,927

The number 2,047 can be written 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 - 1, or 2^11 - 1. It was first factorised in 1536. The number 2,147,483,647, which happens to be 2^31 - 1, was shown to be prime in 1732 by the great German mathematician, Euler. And the first factorization of 147,573,952,589,676,927 (which is 2^67 - 1) was the subject of a paper to the American Mathematical Society in October 1903. These monumental tasks can be carried out by the Factorization program in less than a second!

You can try some even larger whole numbers if you like. It is not necessary to type strings of digits, because the program will accept numbers such as 2^101 - 1 or (3^25-1)*(2^101+1) without breaking sweat. If you want to give it a bit of a workout, try 2^257-1. On my computer, it takes seconds to factorize this 78 digit number.

### Perfect numbers

A perfect number is a whole number which is equal to the sum of its positive factors, other than itself. For example:
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

People have been fascinated by perfect numbers for a very long time. Although the four perfect numbers listed above were the only ones he knew about, Euclid in 300BC showed that they all took a similar form:
6 = 2 * 3 = 2^1 * (2^2-1)
28 = 4 * 7 = 2^2 * (2^3-1)
496 = 16 * 31 = 2^4 * (2^5-1)
8128 = 64 * 127 = 2^6 * (2^7-1)

Euclid even proved that, if the expression in brackets (2^n-1) was a prime number, then 2^(n-1)*(2^n-1) would always be a perfect number. Numbers like 2^n-1 are called Mersenne numbers, after the 17th century monk who studied these numbers. If they cannot be factorized, they are called Mersenne primes. To find perfect numbers, you "only" have to find Mersenne primes. Easier said than done! By the end of the 18th century, two thousand years after Euclid, only four more Mersenne primes had been found. They were 2^13-1, 2^17-1, 2^19-1 and 2^31-1. The corresponding perfect numbers are:
33,550,336 = 4096 * 8191 = 2^12 * (2^13-1)
8,589,869,056 = 65,536 * 131,071 = 2^16 * (2^17-1)
137,438,691,328 = 262,144 * 524,287 = 2^18 * (2^19-1)
2,305,843,008,139,952,128 = 1,073,741,824 * 2,147,483,647 = 2^30 * (2^31-1)

Since the invention of the computer, things have speeded up a bit. Although it took 150 years to find the next Mersenne prime, you can find it in minutes using the Factorization program. Try factorizing 2^n-1 for different values of n until you find one which is prime. Hint: It helps if n itself is prime e.g. 37, 41, 43, 47, 53, 59, 61, 67 etc. Even now, although thousands of computers are engaged in an organised search, only forty three Mersenne primes (and hence forty three perfect numbers) have so far been found.

If you would like to check your answers, or know more about perfect numbers, I strongly recommend the very readable article on The history of perfect numbers from the St Andrews University website. It is a history littered with claims subsequently proved to be false. More advanced students may like to visit Mersenne primes for some background theory.