--On 17. mai 2004 10:13 -0700 p k <[EMAIL PROTECTED]> wrote:
In layman's terms there are infinite number of numbers therefore there are and infinite number of primes. I think it safe to say that since there are infinite number of primes that are infinite number that will take the form of 2^n - 1 granted they are few and far between. I sure there is a way to represent this mathematically.
the proof is more complex, but still rather simple, as proofs go, and is a classical example of "proving a negative":
ASSUME that the number of primes is finite.
THEN IN THEORY, one could list them all, forming "the set of all primes". Then, we can multiply them together, and add 1 to the result.
The resulting number is obviously (by rules of mathematics):
- Not any of the known primes, since it is larger than them all
- Not divisible by any of the known primes, since it is 1 larger than their product
THEREFORE, it is either a prime number which is not a known prime or a product of primes that are not in the set of all known primes.
This is inconsistent with our assumption, which we started with.
THEREFORE, we have proved a negative: The statement "The number of primes is finite" is false.
Take care,
Harald
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