Definition:Euclid Number
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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:
| \(\ds E_0 \ \ \) | \(\ds = p_0\# + 1\) | \(=\) | \(\ds 1 + 1\) | \(\ds = 2\) | ||||||||||
| \(\ds E_1 \ \ \) | \(\ds = p_1\# + 1\) | \(=\) | \(\ds 2 + 1\) | \(\ds = 3\) | ||||||||||
| \(\ds E_2 \ \ \) | \(\ds = p_2\# + 1\) | \(=\) | \(\ds 2 \times 3 + 1\) | \(\ds = 7\) | ||||||||||
| \(\ds E_3 \ \ \) | \(\ds = p_3\# + 1\) | \(=\) | \(\ds 2 \times 3 \times 5 + 1\) | \(\ds = 31\) | ||||||||||
| \(\ds E_4 \ \ \) | \(\ds = p_4\# + 1\) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 + 1\) | \(\ds = 211\) | ||||||||||
| \(\ds E_5 \ \ \) | \(\ds = p_5\# + 1\) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 + 1\) | \(\ds = 2311\) | ||||||||||
| \(\ds E_6 \ \ \) | \(\ds = p_6\# + 1\) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1\) | \(\ds = 30031\) |
Also see
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.
Sources
- Weisstein, Eric W. "Euclid Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EuclidNumber.html