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An integer greater than one is called a
prime number if
its only positive divisors (factors) are one and itself.
For example, the prime
divisors of 10 are 2 and 5, and the first six
primes are 2, 3, 5, 7, 11, and 13.
By the fundamental theorem of arithmetic we know that all
positive integers factor
uniquely into a product of primes.
Technical comment on the definition:
In the integers we can easily prove the following
When we study other number systems, these properties may not hold.
So in these systems of integers (often called rings) we often make the
- A positive integer
p, not one, is prime if whenever it divides the product of integers
ab, then it divides a or b (perhaps both).
- A positive
integer p, not one, is prime if it can not be decomposed into factors
p=ab, neither of which is 1 or -1.
- Any element which divides one is a
- An element p, not a unit, is prime if
whenever it divides the product of integers ab, then it divides
a or b (perhaps both).
- An element p, nonzero
and not a unit, is called irreducible if it can not be
decomposed into factors p=ab, neither of which is a
See Also: PrimeNumberThm, PrimeGaps
Related pages (outside of this work)