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 letâ€™s consider it composite. This means that it must be divisible (without remainder) by prime numbers. However, none of the primes up to p will divide n (since we would always have remainder 1), so any number which does divide n must be greater than p. This means that there is a prime number greater than p after all. Thus, whether n is prime or composite, our supposition that there is a largest prime number is false. Therefore, the set of prime numbers is infinite.

Notes:

There is always one larger.

Folksonomies: mathematics theorem prime numbers proof

Taxonomies:
/science/mathematics/arithmetic (0.706196)
/science/computer science/cryptography (0.472902)
/religion and spirituality (0.447503)

Keywords:
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Concepts:
Prime number (0.963249): dbpedia | freebase | opencyc
Mathematics (0.868397): dbpedia | freebase | opencyc
Natural number (0.680306): dbpedia | freebase | opencyc
Prime numbers (0.570217): dbpedia
Mersenne prime (0.569674): dbpedia | freebase | yago
Number theory (0.521586): dbpedia | freebase | opencyc
Great Internet Mersenne Prime Search (0.505930): dbpedia | freebase | yago
Largest known prime number (0.487893): dbpedia | freebase

Philosophy of Mathematics
Books, Brochures, and Chapters>Book:  Brown, James Robert (2012-10-12), Philosophy of Mathematics, Routledge, Retrieved on 2013-06-18