Definition:Euclid Number

Definition

A Euclid number is a natural number of the form:

$E_n := p_n\# + 1$

where $p_n\#$ is the primorial of the $n$th prime number.

Sequence of Euclid Numbers

The sequence of Euclid numbers begins as follows:

 $\displaystyle E_0 \ \$ $\displaystyle = p_0\# + 1$ $=$ $\displaystyle 1 + 1$ $\displaystyle = 2$ $\displaystyle E_1 \ \$ $\displaystyle = p_1\# + 1$ $=$ $\displaystyle 2 + 1$ $\displaystyle = 3$ $\displaystyle E_2 \ \$ $\displaystyle = p_2\# + 1$ $=$ $\displaystyle 2 \times 3 + 1$ $\displaystyle = 7$ $\displaystyle E_3 \ \$ $\displaystyle = p_3\# + 1$ $=$ $\displaystyle 2 \times 3 \times 5 + 1$ $\displaystyle = 31$ $\displaystyle E_4 \ \$ $\displaystyle = p_4\# + 1$ $=$ $\displaystyle 2 \times 3 \times 5 \times 7 + 1$ $\displaystyle = 211$ $\displaystyle E_5 \ \$ $\displaystyle = p_5\# + 1$ $=$ $\displaystyle 2 \times 3 \times 5 \times 7 \times 11 + 1$ $\displaystyle = 2311$ $\displaystyle E_6 \ \$ $\displaystyle = p_6\# + 1$ $=$ $\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1$ $\displaystyle = 30031$

Source of Name

This entry was named for Euclid.

Historical Note

The name Euclid number derives from Euclid's proof of the Infinitude of Prime Numbers.

It is often stated (erroneously) that this proof relies on these numbers.

However, recall that Euclid did not begin by assuming that the set of all primes is finite. His proof, found in Euclid's The Elements as Proposition $20$ of Book $\text{IX}$: Euclid's Theorem, proceeded as follows:

Take any finite set of primes (it could be any set, for example $\left\{ {3, 211, 65537}\right\}$). Then it follows that at least one prime exists that is not in that set.

However, the numbers themselves are mildly interesting in their own right.

Hence, in honour of Euclid, they have been (however mistakenly) named after him.