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I just see that in this entry it said
Classically, 1 was also counted as a prime number, …
If this is really true, it would be good to see a historic reference. But I’d rather the entry wouldn’t push this, since it seems misguided and, judging from web discussion one sees, is a tar pit for laymen to fall into.
The sentence continued with
…$[$ the number 1 is $]$ too prime to be prime.
and that does seem like a nice point to make. So I have edited the entry to now read as follows, but please everyone feel invited to have a go at it:
A prime number is a natural number which cannot be written as a product of two smaller numbers, hence a natural number greater than 1, which is divisible only by 1 and by itself.
This means that every natural number $n \in \mathbb{N}$ is, up to re-ordering of factors, uniquely expressed as a product of a tuple of prime numbers:
$n \;=\; 2^{n_1} 3^{n_2} 5^{n_3} 7^{n_4} 11^{ n_5 } \cdots$This is called the prime factorization of $n$.
Notice that while the number $1 \in \mathbb{N}$ is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in mathematics of objects being too simple to be simple one might say that 1 is “too prime to be prime”.
After the comment on primes as ideals, I added the following remark on how that leads over to the dual perspective of number theory as arithmetic geometry:
From the Isbell-dual point of view where a commutative ring, such as the integers $\mathbb{Z}$ is regarded as the ring of functions on some variety, namely on Spec(Z), the fact that prime numbers correspond to maximal ideals means that they correspond to the points in this variety, one also writes
$(p) \in Spec(\mathbb{Z}) \,.$This dual perspective on number theory as being the geometry (algebraic geometry) over Spec(Z) is called arithmetic geometry.
In ancient Greece, if that’s what was intended by ’classically’, 1 was largely not considered as a ’number’, let alone a prime number.
…they correspond to the points in this variety…
sounds like all points, when there’s also $(0)$.
Perhaps drop ’the’, or add ’closed’.
Thanks, right. Changed it to “closed points”.
For completeness, I have added the pointer at closed point here.
The consideration of $1$ as a prime continued into the $20^{th}$ century, at least for some like D.H. Lehmer who famously compiled a list of primes. See page 9 here for evidence.
Thanks, Todd, I have added that pointer here.
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