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> 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 exclud

> 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.
    - dr_dshiv 8y ago
      Some [0] have claimed that the Pythagoreans didn't consider 1 and 2 to be numbers, proper. This helps explain why. [0] http://www.math.tamu.edu/~dallen/history/pythag/pythag.html http://www.math.tamu.edu/~dallen/history/pythag/pythag.html
      - nyc111 8y ago
        The referenced article mention this in passing without giving a reason. But didn't Pythgoras use musical whole number ratios like 2:1, 3:2 and 4:3?If so he must have considered 1 and 2 to be numbers.
        dr_dshiv 8y ago
        Sorry to be lazy about citation. Here [1] is a peer reviewed article with better internal citations about Pythagorean mathematical perspectives on 1 and 2. It has been argued [2] that the Pythagorean numbers were quite different from numbers as we think of them in arithmetic. The One (or Oneness), for instance, is more than the number one. However, because these views extended to at least 10, it isn't an argument for disincluding 1 & 2. (BTW, [2] is probably the best thing I've ever downloaded from Kindle. Highly recommended. Be sure to read introduction.) [1] Caldwell, C. K., & Xiong, Y. (2012). What is the smallest prime?. Journal of Integer Sequences, 15(2), 3. [2] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
        nyc111 8y ago
        Thanks for the references. I could only find this online which looks similar to what you have by the same authors: https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.p... Looks very interesting. Reading it now.
        dr_dshiv 8y ago
  - skh 8y ago
    Adding to what amelius said. The formal term for the algebraic system we are talking about is ring. We want to consider all the elements of the ring to be numbers since they are all part of the same algebraic system. Also, let’s broaden the perspective a bit and think about rational numbers. These include all integers and within the system of rational numbers every nonzero element is a unit. So 2 which is prime within the ring of integers is no longer prime within the ring of rational numbers. Every rational number is a multiple of 2 in the sense of 2*(a/2) = a The notion of primality is not a property of a number it is a property of a number within the structure of a ring. We don’t want to think of units as not being numbers. Also, the construction of natural numbers is such that 1 is the successor of 0. It just happens that when considering the operation of multiplication 1 is a unit but this is happenstance and not a reason to exclude 1 as a number. It’s worth noting that under addition 0 is a “unit” (really identity but plays additive role that 1 does under multiplication).
tzs 8y ago
To add to what others said about it being somewhat arbitrary, it should be noted that in different time and places other definitions have been used. Each of the following sets has been "the primes" to mathematicians at some time/place: P1 = {1, 2, 3, 5, 7, 11, ...} P2 = {2, 3, 5, 7, 11, ...} P3 = {3, 5, 7, 11, ...} The underlying mathematics is the same no matter which of those you call "the primes". All that really changes is what you then have to say when you want a specific one of those sets. If P1 is "the primes", and your theorem needs a p that is a member of a specific one of those sets, you have to say "Let p be...": P1: "...a prime" P2: "...a prime other than 1" P3: "...an odd prime" If P2 is "the primes", it is: P1: "...1 or a prime" P2: "...a prime" P3: "...an odd prime" If P3 is "the primes" for you, it is: P1: "...1, 2, or a prime" P2: "...2 or a prime" P3: "...a prime" Given the state of mathematics since the 19th century, P2 as the "primes" probably results in the minimum verbiage. I don't think anyone has used P3 in a very long time, so unless you are studying ancient math history you probably will never encounter anything using it. P1 and P2 were used together up until at least the 18th century, with some mathematicians using P1 even longer, but pretty much everything you will encounter now will use P2. Even is a book you might read now says it is presenting the historical proof of some theorem as it was originally proven, and that proof has done by someone who used P1 as "the primes" in the proof, the book will almost certainly reword it to be for P2 prime. You will probably only ever have a chance of running into P1 primes if you actually go to original sources, finding copies the actual books or articles or papers of mathematicians from back when many used P1 primes.

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.
  - dr_dshiv 8y ago
    Some [0] have claimed that the Pythagoreans didn't consider 1 and 2 to be numbers, proper. This helps explain why. [0] http://www.math.tamu.edu/~dallen/history/pythag/pythag.html http://www.math.tamu.edu/~dallen/history/pythag/pythag.html
    - nyc111 8y ago
      The referenced article mention this in passing without giving a reason. But didn't Pythgoras use musical whole number ratios like 2:1, 3:2 and 4:3?If so he must have considered 1 and 2 to be numbers.
      - dr_dshiv 8y ago
        Sorry to be lazy about citation. Here [1] is a peer reviewed article with better internal citations about Pythagorean mathematical perspectives on 1 and 2. It has been argued [2] that the Pythagorean numbers were quite different from numbers as we think of them in arithmetic. The One (or Oneness), for instance, is more than the number one. However, because these views extended to at least 10, it isn't an argument for disincluding 1 & 2. (BTW, [2] is probably the best thing I've ever downloaded from Kindle. Highly recommended. Be sure to read introduction.) [1] Caldwell, C. K., & Xiong, Y. (2012). What is the smallest prime?. Journal of Integer Sequences, 15(2), 3. [2] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
        nyc111 8y ago
        Thanks for the references. I could only find this online which looks similar to what you have by the same authors: https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.p... Looks very interesting. Reading it now.
        dr_dshiv 8y ago
- skh 8y ago
  Adding to what amelius said. The formal term for the algebraic system we are talking about is ring. We want to consider all the elements of the ring to be numbers since they are all part of the same algebraic system. Also, let’s broaden the perspective a bit and think about rational numbers. These include all integers and within the system of rational numbers every nonzero element is a unit. So 2 which is prime within the ring of integers is no longer prime within the ring of rational numbers. Every rational number is a multiple of 2 in the sense of 2*(a/2) = a The notion of primality is not a property of a number it is a property of a number within the structure of a ring. We don’t want to think of units as not being numbers. Also, the construction of natural numbers is such that 1 is the successor of 0. It just happens that when considering the operation of multiplication 1 is a unit but this is happenstance and not a reason to exclude 1 as a number. It’s worth noting that under addition 0 is a “unit” (really identity but plays additive role that 1 does under multiplication).

tzs 8y ago

To add to what others said about it being somewhat arbitrary, it should be noted that in different time and places other definitions have been used. Each of the following sets has been "the primes" to mathematicians at some time/place: P1 = {1, 2, 3, 5, 7, 11, ...} P2 = {2, 3, 5, 7, 11, ...} P3 = {3, 5, 7, 11, ...} The underlying mathematics is the same no matter which of those you call "the primes". All that really changes is what you then have to say when you want a specific one of those sets. If P1 is "the primes", and your theorem needs a p that is a member of a specific one of those sets, you have to say "Let p be...": P1: "...a prime" P2: "...a prime other than 1" P3: "...an odd prime" If P2 is "the primes", it is: P1: "...1 or a prime" P2: "...a prime" P3: "...an odd prime" If P3 is "the primes" for you, it is: P1: "...1, 2, or a prime" P2: "...2 or a prime" P3: "...a prime" Given the state of mathematics since the 19th century, P2 as the "primes" probably results in the minimum verbiage. I don't think anyone has used P3 in a very long time, so unless you are studying ancient math history you probably will never encounter anything using it. P1 and P2 were used together up until at least the 18th century, with some mathematicians using P1 even longer, but pretty much everything you will encounter now will use P2. Even is a book you might read now says it is presenting the historical proof of some theorem as it was originally proven, and that proof has done by someone who used P1 as "the primes" in the proof, the book will almost certainly reword it to be for P2 prime. You will probably only ever have a chance of running into P1 primes if you actually go to original sources, finding copies the actual books or articles or papers of mathematicians from back when many used P1 primes.