My strong opinion of why 1 is a prime number.

[acarli]

Firstly, I know this is a silly argument about semantics. BUT.. it still grinds my gears everytime I read an article about prime numbers and not mentioning 1 as a prime number.

I have an argument for 1 actually being a prime number.

Wikipedia's definition of a prime number is:
https://en.wikipedia.org/wiki/Prime_number

"A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers."

ok, they say greater than 1, right off the bat. Why? seems arbitrary to me.

Here's another definition:
https://www.mathopenref.com/prime-number.html

"A prime number is a positive integer that has no integer factors except one and itself. "

https://www.mathopenref.com/prime-number.html
Actually, "1" does fit into this definition, at least from an engineering perspective. So "1" by definition does fit. "1" is divisible aby itself, and only no integer factors except itself. Both conditions met.

Definition.com, I like this website, they give da few definitions.

https://www.dictionary.com/browse/prime-number

"positive integer that is not divisible without remainder by any integer except itself and 1, with 1 often excluded: "

"often excluded", leaves some wiggle room for interpretation.

I think that the main divide in this silly argument is whether or not there have to be two discreet positive divisors. I see these things combined with "common core" math, and role my eyes.

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[qwk]

I have an argument for 1 actually being a prime number.

No, you don't

Wikipedia's definition of a prime number is:
[...]

That's just a dumbed-down definition for the layman.
Which is what wikipedia is for, so no reason to criticize them for it.

Here's another definition:
https://www.mathopenref.com/prime-number.html

"A prime number is a positive integer that has no integer factors except one and itself. "

https://www.mathopenref.com/prime-number.html
Actually, "1" does fit into this definition, at least from an engineering perspective.

And here's your mistake.
An engineering perspective is irrelevant to a mathematical definition.

As an engineer, what would you say if I told you that e.g. I find the definition of a "nut" doesn't fit my perspective as a cook?

[acarli]

No, you don't
......
An engineering perspective is irrelevant to a mathematical definition.

As an engineer, what would you say if I told you that e.g. I find the definition of a "nut" doesn't fit my perspective as a cook?

Upps. I got a fish on the line, lol
ok, but what if the definition is arbitrary. Arbitrary nonsense doesn's add value to the definition. It just makes it more complicated, by adding a rule that doesn't make any difference.
For example. Here's an example. Does 25cm four times fit in 1 m? (Pssst.. the answer is "yes") Does 100cm fit in 1m? More importantly, does 1 m fit into 1m?

[qwk]

ok, but what if the definition is arbitrary.

A definition is by its very nature arbitrary, don't you agree?
If it's not arbitrary, it's no longer a definition, but something that can be deduced from prior assumptions / definitions.

More importantly, does 1 m fit into 1m?

Depending on who you ask.
If it's about a mathematical set of values, 1m is not part of that set (actually, not being a mathematician, I'm a little unsure about this), while 100cm is.
An engineer would probably say 1m fits into 1m, but I'm no engineer, either.

[Foxpup]

An engineer would probably say 1m fits into 1m, but I'm no engineer, either.

Actually an engineer would be highly suspicious of a measurement with but a single significant figure and no specified tolerance. OP (and, to be fair, common core) seems to forget that numbers are not arbitrary things that can be defined however we want, instead they have meaning according to their real-world use, and the definitions must match this true meaning, even if it requires making exceptions for exceptional numbers like 0 and 1. Many theorems involving prime numbers simply don't work if 1 is considered to be prime, so for the sake of math actually working instead of being a meaningless bunch of scribbles on a blackboard, the definition of primality must exclude the number 1.

[acarli]

An engineer would probably say 1m fits into 1m, but I'm no engineer, either.

Actually an engineer would be highly suspicious of a measurement with but a single significant figure and no specified tolerance. OP (and, to be fair, common core) seems to forget that numbers are not arbitrary things that can be defined however we want, instead they have meaning according to their real-world use, and the definitions must match this true meaning, even if it requires making exceptions for exceptional numbers like 0 and 1. Many theorems involving prime numbers simply don't work if 1 is considered to be prime, so for the sake of math actually working instead of being a meaningless bunch of scribbles on a blackboard, the definition of primality must exclude the number 1.

ok, I appreciate your feedback. here's an explanation for 6 year olds of why 1 isn't considered prime due to theorems complications. For me, the most compelling argument is that if we do call 1 a prime, then we will have to append a phrase "ALL except for 1" in a lot of theorems that use prime numbers as solutions.

http://mathforum.org/library/drmath/view/57058.html

I am close enough to see your point. I say "close enough" because when you define factors of 1, they are infinite, but we don't treat them as infinite from a commons sense point of view. ir. 1*1, 1^76 etc.

Even in this reasoning, you could simply add 'DISTINCT' clause. I opened this thread with a "Change my mind" attitude.
Valid points, and I learned something.

[SaltySpitoon]

1 isn't "really" a number. Its a standard basis for numbers. When you say that three is a prime number, you are saying that 3 * the unit (1) = 3. Three cannot be divided by any integer besides the unit. Saying the unit * the unit = 1 isn't interesting, useful, and is up for debate.

I can find you a proof that explains that 1 * 1 is not the original value of 1, but there are a handful of assumptions there that I'm not sure you are making.