# 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:**

prime numbers (0.998950 (negative:-0.411190)), largest prime number (0.774657 (negative:-0.421883)), primes (0.683002 (negative:-0.196899)), original supposition (0.656978 (negative:-0.221876)), finite number (0.612381 (negative:-0.349956)), theorem (0.577841 (negative:-0.521906)), Proof (0.571470 (negative:-0.469031)), remainder (0.563827 (neutral:0.000000)), set (0.541033 (neutral:0.000000)), Infinite (0.466623 (neutral:0.000000))

**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

**Folksonomies:**mathematics philosophy