Euclid's Theorem/Fallacy
Jump to navigation
Jump to search
Fallacy
There is a fallacy associated with Euclid's Theorem.
It is often seen to be stated that: the number made by multiplying all the primes together and adding $1$ is not divisible by any members of that set.
So it is not divisible by any primes and is therefore itself prime.
That is, sometimes readers think that if $P$ is the product of the first $n$ primes then $P + 1$ is itself prime.
This is not the case.
For example:
- $\left({2 \times 3 \times 5 \times 7 \times 11 \times 13}\right) + 1 = 30\ 031 = 59 \times 509$
both of which are prime, but, take note, not in that list of six primes that were multiplied together to get $30\ 030$ in the first place.