## The economy of thought

• 64
As conciseness is one of the main mathematical features, I would like to discuss one particular instance of it. Can someone please summarize in that context the usefulness of excluding number one from the set of prime numbers? As the definition of prime numbers would be more concise without it, ie if one was included, and in fact it was at the beginning, first great contributors to number theory who laid foundations to prime number theory considered it to be prime, exclusion was introduced later, without much change in the essence of the theory, so it must have payed off somehow in terms of development of shorter expressions of consequences of somewhat longer definition, and I would like to know all places where it showed to be the case. So, the definition is, for those who are really unfamiliar with the topic: prime numbers are natural numbers that are divisible by exactly two distinct divisors, by one and by themselves. The definition that would include one, would be like this: prime numbers are natural numbers that are divisible only by one and by themselves. Note that further shortening of the definition by omitting the crucial condition “only by” would be a blunder, since all natural numbers are divisible by one and by themselves, which would of course make the definition pointless.
I know there are already many answers online to the question “why 1 is not prime”, such as https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/ for example, but I didn’t find a satisfactory one.
• 65

Marta and María are sisters. Marta has two nieces who are not Maria's nieces. It's possible?
• 135
Marta and María are sisters. Marta has two nieces who are not Maria's nieces. It's possible?
María
Maria

• 945

The number one is being used as the only divisor of a prime. It is difficult to understand how the function can be applied to the value given this job.
• 65

The definition of a prime number is Pythagorean. For the Pythagoreans the number 1 was not even, nor odd, but even-odd. It was the number that was "outside" the numbering system. That is why in the definition of prime it is not introduced as an element of primality. Nor is 0 used, by the way. By dividing it by itself, we get an indeterminacy. However, the Pythagoreans did not have a concept of 0 and, therefore, it is not included in the definition of a prime number, just as we do not introduce irrational numbers.

I insist: "Martha and Mary are sisters. Marta has two nieces who are not Mary's nieces. It's possible?"
• 135
"Martha and Mary are sisters. Marta has two nieces who are not Mary's nieces. It's possible?"

Yes
• 65

Why?
• 135

Marta may be unrelated to Martha / not the same person. And even if she was Martha, the children may be Mary's children. If they are Mary's children, they are not Mary's nieces.
• 8.6k
The reason behind excluding 1 from the set of primes is that we wouldn't/couldn't have a unique prime factorization for a given number:

For example,
9 = 1 × 3 × 3
9 = 1 × 1 × 3 × 3
9 = 1 × 1 ×...× 1 × 3 × 3
and so on...
• 1.8k
Can someone please summarize in that context the usefulness of excluding number one from the set of prime numbers?

Haven't I seen this trolled around the Internet on at least two other forums? The answer in that SciAm article is perfectly satisfactory.

In any event 1 isn't prime because

* Excluding it makes the statement of the fundamental theorem of arithmetic simpler. This is the theorem on unique factorization into products of prime powers, such as $15 = 3 \times 5$ or $12 = 2^2 \times3$. If we included 1 as a prime then $12 = 2^2 \times 3 = 1^2 \times 2^2 \times 3 = 1^3 \times 2^2 \times 3$ etc. So you'd have to say "except for powers of 1" in the statement of the theorem.

The reason behind excluding 1 from the set of primes is that we wouldn't/couldn't have a unique prime factorization for a given number:

Great minds think alike.

* 1 is a unit in the ring of integers and units aren't prime.

* The ideal generated by 1 is not a prime ideal in the ring of integers because a prime ideal must be a proper ideal, and the ideal generated by 1 is not proper.

• 1.8k
I insist: "Martha and Mary are sisters. Marta has two nieces who are not Mary's nieces. It's possible?"

Marta and Maria are nuns in a convent. Or, in American slang, they are African-American women.
• 64
OK, Paul S already answered that Mary does not refer to her own children as her nieces, and he even noticed the missing h in Marta, which I didn’t, so kudos to him for reading precision.
• 64
I think this is more like necroposting than trolling.
• 65

I think that is the simplest solution. Congratulations.

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