Saturday, November 6, 2010

Fermat's Last Theorem - Numbers

The history of number theory decribed in the book "Fermat's Enigma" provided additional excitment to the reading. It described how common number concepts we learn from elementary school to high school, such as prime number, negative number, irrational number, imaginary number, and the concept of infinity, were developed hundreds and thousands of years ago, along with the evolving story of proving Fermat's last theorem. (Note: Fermat's last theorem: The equation x^n+y^n=z^n (*) has no integer solution if n>2. n is an integer)
After the proof for n=3 by Euler, there was no progress in the proof for nearly 75 years until a woman mathematician Sohpie Germain proposed an approach for certain type of prime numbers, such that p and 2p+1 are both primes. For example 2,3,5 are such primes, 7 is not.

Prime Numbers
Prime numbers are those which can be divided only by itself and 1, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, ...7919 (the 1000th prime number) ...., as one can see from the list, there are much less prime numbers than whole numbers. The Fermat's last theorem seemed to be not that daunting any more once ancient mathematicians realized that if they could prove the Fermat's theorem for prime numbers then they proved the theorem because all numbers can be factorized into prime numbers.
Unfortunately there are still infinite prime numbers!! The proof was initially given by the Greek mathematician Euclid of Alexandria, i.e. Euclid. The proof is remarkably easy to understand. Here is the excerpt of the proof.
Assume there are only finite number of prime numbers, p1, p2,, then we can construct a new number q
q=p1 x p2 x p3 x ...x pn +1.
If q is not a prime number, then it must be factorized by prime numbers, bu it can not be factorized by existing prime numbers due to its construction. q must be either factorized by new prime numbers not in the known finite list, or q itself is a prime number. So there are infinite prime numbers!
The method of the proof is called proof by contradiction. It is a powerful method.
Perfect Numbers

Almost opposite to prime numbers, the perfect number is a number which equals to the sum of all of its divisors excluding itself, e.g. 6=1+2+3. The first 4 perfect numbers are
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
There are remarkable less perfect numbers than prime numbers. It is very hard to find perfect numbers, up to 2009, only 47 perfect numbers were found, the 47th perfect number has 25956377 digits! (more than 25 million) - here is the current list
Pythagoras and followers studied perfect number extensively and found many interesting properties they posses, such as all perfect numbers can be written as the sum of consecutive counting numbers, e.g. 8128=1+2+3+4+...+127. Another properties they posses is that all perfect numbers can also be written as 2^n * (2^(n+1)-1), 8129= 2^6 (2^7 -1).
Perfect numbers are perfect in themselves. One claim went as far as saying: "6 is a number perfect in itself, and not because God created all things in 6 days; rather the inverse is true; God created all things in 6 days because this number is perfect."

Finding perfect number is hard, verification of a large perfect number is not easy either. I used excel to verify 8128, it would take a computer program with a smart algorithm to check on the very large perfect numbers. Verification of reasonably large perfect numbers in Excel would be good mental exercises.
Mathematical Revolution due to Power of Logic

Typical thinking is that if a person is logic in nature, he is less inventive because being logic means you follow existing rules. This is not true in the world of math, the the revolutionary concepts in the history of math came from exact logic and deep thinking.

In mathematics, there is need for completeness, which simply says that for a given rule of calculation, however you apply it, you should get a results. The concept of negative number was developed by Hindus because of the need of completeness. Say 5-3=2 was natural for ancient people, what about 3-5? which can not be expressed as a natural number (1,2, 3, 4, ...) - this need of completeness and logic in math led to the development of negative number concept!

Similar story occurred to the discovery of irrational number in ancient Greece. They believed that all numbers were rational. When people pondered what a number would be square root of 2. It was found that no rational number p/q could represent square root of 2, logic demanded the expansion of math concept to irrational number, a revolutionary concept at the time, which caused the life of its first discoverer. The development of the concepts of zero and imaginary number happened in striking similar fashion. Learning how ancient great mathematicians developed the math concepts we take for granted is really exciting. To replicate what they did is fun for amateurs like me. I learned the concept of irrational number in high school, but I never asked why. Now I can easily prove that square root of 2 is irrational using the method proof by contradiction. - it is exhilarating!

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