can you believe this small 500 page book costs $100 – well that is the book i need to buy for my required abstract algebra course for my math major. good thing i ordered the cheapest used one from amazon. with shipping it’s $72, still cheaper than the used book price at my university book store (without tax either). after i get that one, i will return the one i purchased today at school.
while i was waiting for another class, i read the first chapter of that book and actually found a couple of examples extremely amusing. *geek alert*
basically the first chapter is validating why testing can not prove math. so they gave an example. it’s a call Pell’s equation – m^2 = p*n^2+1, where p is a prime and there is a way of calculating all possible solutions of it. A spectacular example of Pell’s equation involves the prime p = 1,000,099; the smallest n for which 1,000,099n^2+1 is a perfect square has 1116 digits. THAT IS CRAZY. it’s saying you can test and test, even if for ages you can not find a contradiction to a statement, that still proves nothing. coz you never know when a contradition will occur. in that particular example the first contradiction comes when a number has 1116 digits.
another example: Gold-bach’s conjecture: every even number m >= 4 is a sum of two primes. No one has ever found a counterexample to Gold-bach’s conjecture, but neither has anyone ever proved it. at present, the conjecture has been verified for all even numbers m < 10^13 by H.J.J. te Riele and J.-M. Deshouillers. It has been proved by J.-R. Chen that every sufficiently large even number m can be written as p + q, where p is prime and q is "almost" a prime; that is, q is either a prime or a product of two primes. Even with all of this positive evidence, however, no mathematician will say that Gold-bach's conjecture must, therefore, be true for all even m.