11–20 of 33 posts
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
Edit: I corrected something in the original comment. > There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. We can generalize this. There are 2 kinds of primes with respect to mod 10. Those whose modulo is 2 or 5, and those whose mod x falls into some other odd number. Of the latter numbers, the more prim…
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
In this particular split of the prime numbers there are 3 categories, and 2 is the only member of one of the categories, while the rest are evenly divided. So it's a solitary outlier.
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
1 is a unit.
It can turn up unexpectedly. For instance, here's a problem/puzzle that a co-worker came up with: Find all positive integers b for which whenever (x^2 + xy + y^2) written in base b (for some integers x and y) ends with 0, it ends with two 0s. (Answer in rot13: gubfr gung ner n cebqhpg bs qvfgvapg cevzrf gung ner rnpu gjb zbqhyb guerr. This can be proved using quadratic reciprocity, at least the special case that (-3) is a square mod p iff p is a square mod 3.)
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
In particular, the papers "What is the smallest prime?" https://arxiv.org/abs/1209.2007 and "The History of the Primality of One: A Selection of Sources" https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... have good history on the matter (e.g. 1, 2, and 3 were all answers at various points — note that at some point in history 1 wasn't even considered a number).
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
- A prime number is a positive prime element of Z
- A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both)
- An element which generates a prime ideal is a prime element
- A ring divided by a prime ideal is an integral domain
- All integral domains have a field of fractions
- The zero ring is not a field
Make 1 a prime at the first step and things go wrong at the final step. Most of these definitions have a good argument, except the first arbitrarily excludes the non-positive numbers. This includes 0 which is a perfectly acceptable prime according to all other definitions (and unlike the negative primes, is not simply the negative of a prime number).
Edit: I corrected something in the original comment. > There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. We can generalize this. There are 2 kinds of primes with respect to mod 10. Those whose modulo is 2 or 5, and those whose mod x falls into some other odd number. Of the latter numbers, the more prim…
but what puzzles me is that 11,13,17,19 are primes just like 101,103,107,109
but the real punchline is that 23 is the next prime after 19 AND 113 is the next prime after 109. WHY!?
this must be connected to 2*5
but I only have questions and confusion
Edit: I corrected something in the original comment. > There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. We can generalize this. There are 2 kinds of primes with respect to mod 10. Those whose modulo is 2 or 5, and those whose mod x falls into some other odd number. Of the latter numbers, the more prim…
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
• When you work modulo 4 (as in the article in the context where this comes up), there are three kinds of primes: 2 is a solitary outlier, and all other primes are equally distributed between being 1 and 3 mod 4.
• When you work modulo 10, there are six kinds of primes: 2 and 5 are outliers, and all other primes are equally distributed between being 1, 3, 7 and 9 mod 10.
So 2 is an outlier when working mod 4 (or any even number). Mod 4 is crucially important in the context of quadratic reciprocity, so it matters here.
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
Earlier quoted context omitted.
Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.
And while we are at it why is 1 not a prime? It conforms to the definition.
But it is part of a general pattern that certain objects are "too simple to be simple" [0].
Here "simple" should be thought of as atomic, indecompossible, without smaller parts.
- prime numbers are simple wrt multiplication, 1 is too simple to be prime
- the zero ring is too simple to be a field
- the trivial group is too simple to be a simple group
- the empty topological space is not connected
- etc...
It turns out that if one adheres to this convention, then theorem statements generally become shorter, and have fewer side conditions.