Thursday, April 19, 2007

Proofs

I want to learn math and computer science. I want to meet others interested in learning these topics. So, I'm starting a blog in hopes of making friends. The whole point of this blog for me is to make friends and to have fun.

Right now I am reading How To Prove It by Daniel J. Velleman, and I'm on the introduction. I'm trying to work a bit each day on doing proofs, mostly I'm working on proofs while riding the CTA train in chicago. Well, if I fall asleep so be it, but i first try to work on proofs.

The first chapter has some definitions:
1. prime numbers - can not be written as the product of two smaller positive integers
2. conjecture - a guess, an educated guess
3. Theorem - a conjecture that has been proven
4. mersenne primes - prime numbers of the form 2^n-1
5. twin primes - primes that are two away from each other (examples: 5,7 and 29,31)
6. perfect numbers - n is perfect if n equals the sum of the smaller positive integers that divide n (examples: 6 because 1+2+3 = 6 and 28 because 1+2+4+7+14=28)

I'm going to think about this Conjecture in the book and try to prove it myself before reading the answer:
Conjecture 2: Suppose n is an integer larger than 1 and n is not prime. Then 2^n-1 is not prime.

I'm also going to think about this Theorem from the book and try to prove it myself before reading the anser:
Theorem 3: There are infinitely many prime numbers.

I really have no idea how to prove these two things. Without looking at the book it is hard to know where to even begin thinking of these two problems.

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