Definition:Euclid Number/Historical Note

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Historical Note on Euclid Number

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.