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.

Examples

The first few Euclid numbers are as follows:

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

This sequence is A006862 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Comment

The name derives (erroneously) from Euclid's proof of the Infinitude of Prime Numbers. It is often said that this proof relies on these numbers.

Actually, Euclid did not begin by assuming that the set of all primes is finite. What he did say was: take any finite set of primes (it could be any set, e.g. $\left\{{3, 211, 65537}\right\}$). 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, and hence in honour of Euclid, have been (however mistakenly) named after him.