There's a nice characterization of sums of two squares in terms of their prime factorization), namely that all primes of form 4k+3 appear with even multiplicity. From a quick look through OEIS it looks like there are similar characterizations for a^2 + n*b^2 but this is where I tap out on number theory.
It's striking how complicated quadratic forms are, given how simple they sound. You could probably explain the idea of a quadratic form over a ring to a good high school class, while explaining the proof of the Milnor conjecture would be tough even for graduate students in math.
It seems to be taken directly from the article, but it doesn't begin at the beginning or end at the end or shed any light on the part of the title most likely to be puzzling ("beyond arithmetic").
If the intention was to help out readers who don't know what a quadratic form is, I think a more helpful piece of advice would be: if you don't already know what a quadratic form is, then it is very unlikely that you will get anything much out of this article.
i started my comment with a double-quote indicating that the content is directly from the article starting right at the beginning of the introduction section (to your point about starting at the beginning).
i went down to Euler’s sums of two squares identity which i think demonstrates clearly that even middle school algebra suffices to get a sense of depth from this work showing that a product can also be seen as a sum in a more sophisticated context (ie beyond arithmetic structure emerges).
i fail to understand your disapproval on my providing additional context from the page to indicate that this is likely a more interesting post to a wider set of readers than most might assume just from the title. i didn’t feel the need to editorialize further because i thought the quote says it all on its own.
Comments that are just copy/paste from articles almost always gets downvoted and/or flagged. People can always read the article in question to get the same material. If you think something is interesting about the quoted bit, it's more helpful to add your commentary or even just a "Hey, look this section pretty much just requires middle school math." so people can understand why you're copy/pasting from the article.
Without that contextual clue, your comment appears to just be noise.
in retrospect, i can see why you might say that, but as noted in my previous reply i selected those paragraphs deliberately though i understand now it would have maybe helped to say a few words about why i thought those paragraphs would be helpful in particular (see my previous comment about middle school algebra).
I came across the Brahmagupta's identity mentioned here recently and thought it was pretty cool. https://en.wikipedia.org/wiki/Brahmagupta%27s_identity
It says - Numbers of the form a^2 + n*b^2 are closed under multiplication.
It follows from matrices of the form [[a, (-n)b], [b, a]] being closed under multiplication, and taking determinants.
In more advanced language: For R a commutative ring (like say, the integers) the following function f is a ring homomorphism
Now take determinants.There's a nice characterization of sums of two squares in terms of their prime factorization), namely that all primes of form 4k+3 appear with even multiplicity. From a quick look through OEIS it looks like there are similar characterizations for a^2 + n*b^2 but this is where I tap out on number theory.
It's striking how complicated quadratic forms are, given how simple they sound. You could probably explain the idea of a quadratic form over a ring to a good high school class, while explaining the proof of the Milnor conjecture would be tough even for graduate students in math.
[flagged]
What is the purpose of this comment?
It seems to be taken directly from the article, but it doesn't begin at the beginning or end at the end or shed any light on the part of the title most likely to be puzzling ("beyond arithmetic").
If the intention was to help out readers who don't know what a quadratic form is, I think a more helpful piece of advice would be: if you don't already know what a quadratic form is, then it is very unlikely that you will get anything much out of this article.
i started my comment with a double-quote indicating that the content is directly from the article starting right at the beginning of the introduction section (to your point about starting at the beginning).
i went down to Euler’s sums of two squares identity which i think demonstrates clearly that even middle school algebra suffices to get a sense of depth from this work showing that a product can also be seen as a sum in a more sophisticated context (ie beyond arithmetic structure emerges).
i fail to understand your disapproval on my providing additional context from the page to indicate that this is likely a more interesting post to a wider set of readers than most might assume just from the title. i didn’t feel the need to editorialize further because i thought the quote says it all on its own.
Comments that are just copy/paste from articles almost always gets downvoted and/or flagged. People can always read the article in question to get the same material. If you think something is interesting about the quoted bit, it's more helpful to add your commentary or even just a "Hey, look this section pretty much just requires middle school math." so people can understand why you're copy/pasting from the article.
Without that contextual clue, your comment appears to just be noise.
ok thanks for explaining, that helps.
> i fail to understand your disapproval on my providing additional context from the page
But this isn't additional context. It's just a few copy-pasted paragraphs.
in retrospect, i can see why you might say that, but as noted in my previous reply i selected those paragraphs deliberately though i understand now it would have maybe helped to say a few words about why i thought those paragraphs would be helpful in particular (see my previous comment about middle school algebra).