What is euclid's theory about prime numbers?
That there are an infinite number of prime numbers.
Before we look an explanation or proof, we must agree on some
points
1. The term number means whole number or integer
2. A prime number is any number that has only 2 factors (1 and
itself).
3. All numbers are either prime or the product of 1 or more
primes; try and find a number that you cannot generate as the
product of primes (e.g. 8 = 2x2x2; 36 = 2x2x3x3).
Now:
If you take any two or more prime numbers and find their product
the resulting number will have the prime numbers used as factors.
However, if you add 1 to the number then the prime numbers you used
to produce this number will now no longer be factors of this new
number.
Example
2,3,5 (first three prime numbers) 2x3x5 = 30
30 +1 =31 - now 2,3 and 5 are not factors as you will always
have a remainder of 1 if you divide by any of the three original
prime factors (2,3 or 5).
If you take all of the known prime numbers and find the product
of all of these prime numbers we get a new number (call it Product
of Primes or PP), PP will have all the know primes as its factors.
If we now add one to PP (PP + 1=N) we will get a number, N, that
will have none of the known primes as a factor. If we say that the
highest value prime number known (that we used to generate PP) is
Pi then N must either be prime or have a prime factor greater than
Pi and thus Pi is not the highest prime number. Therefore there are
an infinite number of prime numbers.