18 JUN 2013 by ideonexus

 Proof That the Set of Prime Numbers is Infinite

Theorem: There are infinitely many prime numbers. Proof: Suppose, contrary to the theorem, that there is only a finite number of primes. Thus, there will be a largest which we can call p. Now define a number n as 1 plus the product of all the primes: n = (2 X 3 X 5 X 7 X 11 X...X p) 1 Is n itself prime or composite? If it is prime then our original supposition is false, since n is larger than the supposed largest prime p. So now let’s consider it composite. This means that it must be div...
  1  notes

There is always one larger.

12 JAN 2012 by ideonexus

 2,305,843,008,139,952,128 is the Greatest Perfect Number ...

230(231-1) ... is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for; as they are merely curious without being useful, it is not likely that any person will attempt to find a number beyond it.
  1  notes

231-1 = 2147483647 is a prime number, and a number on which this number relies. The prime number is the largest value for a 32-bit signed integer. Barlow believed no prime greater than this would be discovered as there was no point.