Quadratic Reciprocity: The connection that changed number theory
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Quadratic Reciprocity: The connection that changed number theory
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Re: Quadratic Reciprocity: The connection that changed number theory
#2Edit:
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 primes you consider the more it looks like a uniform distribution. This is related to the Riemann hypothesis [1].
But for the purposes of this article; we have that the mod is 4.
Re: Quadratic Reciprocity: The connection that changed number theory
#3Gauss died in 1855 and his work still influences modern mathematics. Someone told me while in grad school that in number theory we’re still figuring out stuff that Gauss already knew.
Re: Quadratic Reciprocity: The connection that changed number theory
#4Edit: 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'.
Re: Quadratic Reciprocity: The connection that changed number theory
#5Edit: 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'.
And while we are at it why is 1 not a prime? It conforms to the definition.
Re: Quadratic Reciprocity: The connection that changed number theory
#6Gauss died in 1855 and his work still influences modern mathematics. Someone told me while in grad school that in number theory we’re still figuring out stuff that Gauss already knew.
His motto was "Few, but ripe." I'm sure he had no conception of throwing compute power against the wall and see what sticks.
Re: Quadratic Reciprocity: The connection that changed number theory
#7Edit: 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'.
Because many formulas associated to a prime have a factor of p-1, so it's trivial in the case of p=2. There's also other weird stuff like no negatives and no quadratic formula in characteristic 2.
Re: Quadratic Reciprocity: The connection that changed number theory
#8Earlier 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.
Because primes are about factoring and the number of 1's in a factorization is boundless and immaterial.
Re: Quadratic Reciprocity: The connection that changed number theory
#9Earlier 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.
You can’t uniquely factor if you include units, which in the naturals is 1.
You’d also remove every other prime number, since in general we can say that a prime isn’t divisible by a smaller prime.
That we have to add “except 1” to both the factoring theorem and the definition of prime itself is why we don’t include 1 as a prime.
Re: Quadratic Reciprocity: The connection that changed number theory
#10Earlier 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.
Up until sometime in the 19th century 1 was often considered to be prime.
If it is considered to be a prime, then you end up with a lot of theorems that start "for all primes p != 1" or "for all odd primes != 1".
If it is not considered to be a prime, you end up with other theorems that start "for all primes and 1" or "for all odd primes and 1".
There are more really important theorems in the former set than the latter set and so considering it to not be prime became the convention.