5 ms·

Yeah that comment didn’t make much sense. No one would seriously dispute that 2 is the first prime.

by beervirus 6y ago

klyrs 6y ago
Not so! Some, quite seriously, consider -1 to be prime. John Conway was such a person.
- gerdesj 6y ago
  Mr Conway was (RIP) a genius. If he says -1 is prime ... it's prime. End of.
- gowld 6y ago
  Lol check this out https://www.google.com/search?q=is+-1+prime https://www.google.com/search?q=is+-1+prime More seriously http://mathforum.org/library/drmath/view/55958.html http://mathforum.org/library/drmath/view/55958.html But the link to details is dead.
  - Muskwalker 6y ago
    Internet Archive suggests the original discussion was a Usenet thread; here is the link to Google Groups' version of it: https://groups.google.com/d/topic/geometry.research/7pyFhAAye1E/discussion https://groups.google.com/d/topic/geometry.research/7pyFhAAy...
  - klyrs 6y ago
    > Lol check this out https://www.google.com/search?q=is+-1+prime https://www.google.com/search?q=is+-1+prime I saw that, looking for references... very amusing. Even funnier (to me): https://www.google.com/search?q=is+100000010111000000000000000000000000001+a+prime+number https://www.google.com/search?q=is+1000000101110000000000000...
- mantap 6y ago
  -1 can't be prime because many things that use prime numbers require a positive number. For example, how would you define prime powers? (-1)^2=1... You can certainly give a name to the integers {-1} U P, but maybe it would be better to call them "choice" or "select" numbers.
  - klyrs 6y ago
    As John Conway said (see sibling comment, or e.g., [1]): > Every nonzero rational number has a unique factorization into powers of distinct primes. As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely: -1 * 4 * 1/3 * 5 * 1 ... whereas the factorization of 100/3 is uniquely: 1 * 4 * 1/3 * 5 * 1 ... Where it's true that the representation using primes with exponents is not unique, it is true that the representation using powers of primes is unique. That is, your issue regarding (-1)^2 is that there are infinitely many representations of 1 or -1 having the form (-1)^x, but if you evaluate (-1)^x (x being integral, of course) you'll only get one of two numbers. And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive). [1] http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLecture1.mp4 http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)
gerdesj 6y ago
Have you read GEB (Hofstadter)? I'm not an expert but I am capable of taking the piss. It should be pretty obvious. SS
Someone 6y ago
Because “not prime” isn’t the same as “composite” (1 is neither prime nor composite in the modern view) it isn’t that simple. Some ancient mathematicians such as Aristotle and Plato solved that problem by saying 2 is the first number. Others stated 3 to be the first prime That isn’t holdable once you accept 0 and negative integers to be numbers. https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... states that, for example, Goldbach thought 1 to be prime at some time (in a letter to Euler), as did Legendre, Lebesgue (sometimes), Cayley, Kronecker, Hardy, Lehmer (as the article discussed also says), and the aliens in Carl Sagan’s “Contact”. It also gives fairly recent publications that state 1 is a prime. In the end, whether we consider 1 to be prime is more a choice (just as mathematicians commonly chose to pick 0⁰ = 1) than that it necessarily the case. It just is the better choice (https://en.wikipedia.org/wiki/Prime_number#Primality_of_one https://en.wikipedia.org/wiki/Prime_number#Primality_of_one)
- gerdesj 6y ago
  Hui originally said "Do you start counting from 0 or 1" and I think he was taking the piss, as am I. I'm not sure how we go to this point exactly but I've drunk a lot of wine and will now take stage left.
- beervirus 6y ago
  We’re not talking about whether or not 1 is prime. We’re talking about whether 2 is the first prime or the zeroth prime. No one would seriously say the latter.
  - gerdesj 6y ago
    Can there be a zeroth prime? We are counting things and I think that counting starts at one. Given that a counting always starts at one, because that is what defines counting, then there cannot be a zeroth prime.
    - beervirus 6y ago
      That is exactly my point.
xelxebar 6y ago
Context matters. Hui likely used APL to code up the problem and APL let's one choose between 0- and 1-indexing. Reading APL papers from around the time, it looks common to have explicitly declared your indexing convention up front, so my suspicion is that Hui essentially was responding "I'll code up a bit of APL and get you the answer. What are your conventions?"
emtel 6y ago
That wasn’t in dispute. The question was whether you count 2 as the 0th or 1st prime.
murkle 6y ago
Not true, 1 was considered prime quite recently https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/ https://blogs.scientificamerican.com/roots-of-unity/why-isnt...

Yeah that comment didn’t make much sense. No one would seriously dispute that 2 is the first prime.

klyrs 6y ago

Not so! Some, quite seriously, consider -1 to be prime. John Conway was such a person.

gerdesj 6y ago
Mr Conway was (RIP) a genius. If he says -1 is prime ... it's prime. End of.
gowld 6y ago
Lol check this out https://www.google.com/search?q=is+-1+prime https://www.google.com/search?q=is+-1+prime More seriously http://mathforum.org/library/drmath/view/55958.html http://mathforum.org/library/drmath/view/55958.html But the link to details is dead.
- Muskwalker 6y ago
  Internet Archive suggests the original discussion was a Usenet thread; here is the link to Google Groups' version of it: https://groups.google.com/d/topic/geometry.research/7pyFhAAye1E/discussion https://groups.google.com/d/topic/geometry.research/7pyFhAAy...
- klyrs 6y ago
  > Lol check this out https://www.google.com/search?q=is+-1+prime https://www.google.com/search?q=is+-1+prime I saw that, looking for references... very amusing. Even funnier (to me): https://www.google.com/search?q=is+100000010111000000000000000000000000001+a+prime+number https://www.google.com/search?q=is+1000000101110000000000000...
mantap 6y ago
-1 can't be prime because many things that use prime numbers require a positive number. For example, how would you define prime powers? (-1)^2=1... You can certainly give a name to the integers {-1} U P, but maybe it would be better to call them "choice" or "select" numbers.
- klyrs 6y ago
  As John Conway said (see sibling comment, or e.g., [1]): > Every nonzero rational number has a unique factorization into powers of distinct primes. As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely: -1 * 4 * 1/3 * 5 * 1 ... whereas the factorization of 100/3 is uniquely: 1 * 4 * 1/3 * 5 * 1 ... Where it's true that the representation using primes with exponents is not unique, it is true that the representation using powers of primes is unique. That is, your issue regarding (-1)^2 is that there are infinitely many representations of 1 or -1 having the form (-1)^x, but if you evaluate (-1)^x (x being integral, of course) you'll only get one of two numbers. And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive). [1] http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLecture1.mp4 http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)

gerdesj 6y ago

Have you read GEB (Hofstadter)? I'm not an expert but I am capable of taking the piss. It should be pretty obvious. SS

Someone 6y ago

Because “not prime” isn’t the same as “composite” (1 is neither prime nor composite in the modern view) it isn’t that simple. Some ancient mathematicians such as Aristotle and Plato solved that problem by saying 2 is the first number. Others stated 3 to be the first prime That isn’t holdable once you accept 0 and negative integers to be numbers. https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... states that, for example, Goldbach thought 1 to be prime at some time (in a letter to Euler), as did Legendre, Lebesgue (sometimes), Cayley, Kronecker, Hardy, Lehmer (as the article discussed also says), and the aliens in Carl Sagan’s “Contact”. It also gives fairly recent publications that state 1 is a prime. In the end, whether we consider 1 to be prime is more a choice (just as mathematicians commonly chose to pick 0⁰ = 1) than that it necessarily the case. It just is the better choice (https://en.wikipedia.org/wiki/Prime_number#Primality_of_one https://en.wikipedia.org/wiki/Prime_number#Primality_of_one)

gerdesj 6y ago
Hui originally said "Do you start counting from 0 or 1" and I think he was taking the piss, as am I. I'm not sure how we go to this point exactly but I've drunk a lot of wine and will now take stage left.
beervirus 6y ago
We’re not talking about whether or not 1 is prime. We’re talking about whether 2 is the first prime or the zeroth prime. No one would seriously say the latter.
- gerdesj 6y ago
  Can there be a zeroth prime? We are counting things and I think that counting starts at one. Given that a counting always starts at one, because that is what defines counting, then there cannot be a zeroth prime.
  - beervirus 6y ago
    That is exactly my point.

xelxebar 6y ago

Context matters. Hui likely used APL to code up the problem and APL let's one choose between 0- and 1-indexing. Reading APL papers from around the time, it looks common to have explicitly declared your indexing convention up front, so my suspicion is that Hui essentially was responding "I'll code up a bit of APL and get you the answer. What are your conventions?"

emtel 6y ago

That wasn’t in dispute. The question was whether you count 2 as the 0th or 1st prime.

murkle 6y ago

Not true, 1 was considered prime quite recently https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/ https://blogs.scientificamerican.com/roots-of-unity/why-isnt...