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In case anyone reads the article and wonders why 1 is not a prime number I'll give a brief answer. A prime number has to have the property that not every numbe

by skh 8y ago

In case anyone reads the article and wonders why 1 is not a prime number I'll give a brief answer. A prime number has to have the property that not every number is a multiple of it. In the language of abstract algebra this means that the ideal generated by that number is proper (not the entire set of numbers one is considering). Another reason for discounting 1 as a prime number is that considering 1 to be a prime destroys the uniqueness of prime factorization. For instance, 24 is uniquely, up to order of powers of prime factors: 2^3 times 3 If we allow 1 to be prime then I can write 24 as 1^17 times 2^3 times 3 or I can write it as 1^2 times 2^3 times 3 We lose the property that representations of numbers as products of powers of primes is unique. Thus we have a good reason to discount 1 as being prime and there aren't any good reasons to count it as a prime. The definition given in elementary school is not the correct one. It's a working definition that works well and is correct for all positive integers except 1. In my experience people remember the definition of: "only divisible by itself and 1". Then they ask, "Why isn't 1 considered prime then?" It's because the definition given isn't correct.

doomrobo 8y ago
I like the following definition: p is a prime in a ring R iff the ideal generated by p is a prime ideal of R. Equivalently, p is prime iff p | ab implies p | a or p | b. The only issue with this definition is that you're forced to admit that 0 is prime in integral domains.
- tacomonstrous 8y ago
  It's the same issue though, since you have to specify that prime ideals cannot be the whole ring to avoid 1 being prime.
- vram22 8y ago
  >p is a prime in a ring R That reminds of the topic of groups, rings and fields that I had in either last year of high school or an early year of college (Math.). I thought that was a really cool topic. Not too hard (for the stuff we covered), and still some cool results/theorems in it. Also, they underlie a lot of other areas of mathematics. https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)
chimeracoder 8y ago
> A prime number has to have the property that not every number is a multiple of it. I understand why this observation is true given the definition that excludes 1 from being a prime number, but I don't follow why this is a necessary property of prime numbers (justifying the definition in the first place).
- skh 8y ago
  The definition of a prime number is arbitrary. We could have defined the term “prime number” to mean anything. The question is what concept do we wish to capture with the definition. If we allow numbers with the property that all other numbers in the number system are multiples of it then 1 is a prime number and this destroys uniqueness of prime factorization. Since we want to keep unique factorization it is convenient to exclude numbers that are called units (numbers in which all other numbers are multiples of it). Prime ideals need to be proper subsets of the number system or else we’d have to add the phrase “Let P be a proper prime ideal” to a vast number of theorems. This is inconvenient and allowing units to be prime doesn’t give any benefits. Only headaches so it’s best to just exclude them.
  - nyc111 8y ago
    > it is convenient to exclude numbers that are called units (numbers in which all other numbers are multiples of it). Thanks for this explanation. I used to be puzzled about 1 being not a prime but now I feel better. But can we also propose that 1 should not be considered a number? Because 1 is the unit with which all other numbers are measured.
    - amelius 8y ago
      > But can we also propose that 1 should not be considered a number? That would (again) destroy a lot of useful properties without any benefit. > Because 1 is the unit with which all other numbers are measured. Not sure why this would make you question whether 1 is a number.
      - 8y ago
jhncls 8y ago
Maybe it isn't so intuitive after all? Among the famous advocates of the primesness of 1, are [0] Goldbach, Legendre, Lehmer and Carl Sagan. The main discussion seems to come from 1 not being a composite number, but neither being welcomed in the home of the primes. [0]https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...
akalin 8y ago
I prefer to think about it as there are three special types of numbers: zero, units (invertible elements), and prime numbers, which exclude zero and units. In the integers, the units are +/-1, so that’s why they are not prime. In the rationals, every non-zero element is a unit, which is why there are no prime numbers in the rationals.

by skh 8y ago

doomrobo 8y ago

I like the following definition: p is a prime in a ring R iff the ideal generated by p is a prime ideal of R. Equivalently, p is prime iff p | ab implies p | a or p | b. The only issue with this definition is that you're forced to admit that 0 is prime in integral domains.

tacomonstrous 8y ago
It's the same issue though, since you have to specify that prime ideals cannot be the whole ring to avoid 1 being prime.
vram22 8y ago
>p is a prime in a ring R That reminds of the topic of groups, rings and fields that I had in either last year of high school or an early year of college (Math.). I thought that was a really cool topic. Not too hard (for the stuff we covered), and still some cool results/theorems in it. Also, they underlie a lot of other areas of mathematics. https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)

chimeracoder 8y ago

> A prime number has to have the property that not every number is a multiple of it. I understand why this observation is true given the definition that excludes 1 from being a prime number, but I don't follow why this is a necessary property of prime numbers (justifying the definition in the first place).

skh 8y ago
The definition of a prime number is arbitrary. We could have defined the term “prime number” to mean anything. The question is what concept do we wish to capture with the definition. If we allow numbers with the property that all other numbers in the number system are multiples of it then 1 is a prime number and this destroys uniqueness of prime factorization. Since we want to keep unique factorization it is convenient to exclude numbers that are called units (numbers in which all other numbers are multiples of it). Prime ideals need to be proper subsets of the number system or else we’d have to add the phrase “Let P be a proper prime ideal” to a vast number of theorems. This is inconvenient and allowing units to be prime doesn’t give any benefits. Only headaches so it’s best to just exclude them.
- nyc111 8y ago
  > it is convenient to exclude numbers that are called units (numbers in which all other numbers are multiples of it). Thanks for this explanation. I used to be puzzled about 1 being not a prime but now I feel better. But can we also propose that 1 should not be considered a number? Because 1 is the unit with which all other numbers are measured.
  - amelius 8y ago
    > But can we also propose that 1 should not be considered a number? That would (again) destroy a lot of useful properties without any benefit. > Because 1 is the unit with which all other numbers are measured. Not sure why this would make you question whether 1 is a number.
    - 8y ago

jhncls 8y ago

Maybe it isn't so intuitive after all? Among the famous advocates of the primesness of 1, are [0] Goldbach, Legendre, Lehmer and Carl Sagan. The main discussion seems to come from 1 not being a composite number, but neither being welcomed in the home of the primes. [0]https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...

akalin 8y ago

I prefer to think about it as there are three special types of numbers: zero, units (invertible elements), and prime numbers, which exclude zero and units. In the integers, the units are +/-1, so that’s why they are not prime. In the rationals, every non-zero element is a unit, which is why there are no prime numbers in the rationals.