I have this weird inclination to pass out in this Laz-E boy chair all reclined and stuff with a blanket. I just feel I'm not going to be happy in my bed since lately I've been getting little to no sleep and it's frightfully cold...
Apologies if I sound crazy and you lot are stuck with me as my course is on hold due to the teachers being out of the office (seriously..) and I've downed more coffee and sugar today than I ever have in my entire lifetime so I think I'm going to be up a while..
And yes I have been creeping about on the forum lately... Wallaby knows why. The "math" they have assigned me has knocked a few screws loose. I feel crazier and crazier trying to interpret what on earth THIS means:
If n itself is prime then there is a prime greater than p. |
Suppose n is not prime, then there is a number m such that m > 1, m < n and m divides n exactly. Let q be the smallest such number.
Suppose q is not prime, then there would be a divisor r > 1 of q such that r < n and r divides n exactly, so q would not be the smallest such number. Thus q is prime.
Suppose q <= p, then q is one of the primes in the definition of n above, so n/q would equal an integer plus 1/q, which is impossible, since q divides n exactly. Thus if n is not prime then there is a prime greater than p.
Thus whether or not n is prime there is a prime greater than p.
Thus for any prime number there is a larger prime number, so there are infinitely many prime numbers.
There is a simpler proof, given in 1878 by the eminent mathematician Kummer:
Suppose the number of primes is finite: p1, p2, ..., pk, and let n be the product of all these primes. n - 1 > pk so is not prime, so there is some i such that pi divides n - 1. Since pi also divides n, pi divides n - (n - 1) = 1, which is absurd. Thus the number of primes is infinite.
Oh and I ate steak and cheese pie for dinner. That was lovely..