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Here is a delightful interactive proof of Pythagoras theorem, and a brilliant example of the Internet as an educational resource.
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.
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 + 4064People 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.
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