More on integer valued polynomials: https://cameroncounts.wordpress.com/2017/01/31/polynomials-t...
Well I was curious, but there's a lot there I didn't understand. Apparently I'm good enough at math to do the proofs, but not to write the exercises.
Exercise left to the reader:
Prove 7*n^3 + n is divisible by 2
7*n^3 is even when n is even and odd otherwise.
odd + odd is even, as is even + even.
The easy way of seeing the first part is to do the prime factorization. The 7 doesn't matter since it's prime. If n has a 2 in its factorization it now has 2^3. But if it doesn't have a 2 it won't suddenly acquire one.
All the symbol soup proofs aren't wrong but I don't think they satisfyingly explain the why.
Given p(n) = 7n^3 + n:
If n is even, we can choose some m such that n = 2m, and p(n) = p(2m) = 7 * 8m^3 + 2m = 2 * (7 * 4m^3 + m), which is divisible by 2 since we could factor out the 2 at the start.
If n is odd, similarly we can say n = 2m + 1. p(2m) = 7 * (2m + 1)^3 + (2m + 1) = 56m^3 + 84m^2 + 44m + 8 = 2 * (28m^3 + 42m^2 + 22m + 4), which is also divisible by 2 per the 2 at the start.
7n^3 +n (mod 2) = 1 n^3 + n = n + n = 2*n = 0*n = 0
[stub for offtopicness]
I feel silly saying this, but I wish the author would use more periods and fewer exclamation marks.
Elaine Benes would be proud of their writing...
I was in the pool!
You're the second commenter, so far, to mention exclamation marks. What do they mean to you that would bother you so much to point it out, or anyone for that matter? I haven't even noticed them until I read the comments here on hn.
Not gp, but I feel similarly. For me, I can't help read it with emphasis. As in, the voice in my head gets all fancy in an annoying way. If you imagine someone in person reading it out-loud with exaggerated emphasis, that's what it feels like. Same thing with comic books for me, the sprinkled bolded words in dialog are really grating.
To me it's fairly similar to someone making excessive use of CAPS LOCK. It can be used as a stylistic choice at times, but use it TOO MUCH and it just becomes DISTRACTING.
I DON'T SEE A PROBLEM WITH THIS EITHER! BUT I EMPATHIZE! I GET COMMENTS FROM PEOPLE SAYING THAT I'M SOMEHOW YELLING AT THEM ALL THE TIME BUT I'M ACTUALLY SITTING IN SILENCE, TYPING QUIETLY ON A MEMBRANE KEYBOARD! LOL???
Oh this is nothing. One of my colleagues does that and adds random colour changes, underlines and font face changes. It's like working with a serial killer.
Maybe he was a teenager on IRC in the late 90s or early 00s and decided to never change
Thinking about it I guess MSN messenger and My Space also allowed/encouraged font shenanigans? My memory falters
Ahh. I honestly miss that amount of self-expression, garish as it was. Or rather, I intensely dislike the mono-culture where every vertical video with one-word subtitles looks the same.
Interesting topic, but the writing style is so tedious that it really subtracts from the content as a whole. There's on average one exclamation mark per paragraph. Surely every single thought you write down isn't groundbreaking enough to warrant that.
I beg to differ, I think the writing conveys beautifully, the deeper abstract ideas embedded in what appears to be a simple problem - hence, it captures the essential spirit of what math is about
I like when the author’s personality shines through, and frankly I can’t imagine finding occasional exclamation marks _tedious_ of all things. I just don’t take things so seriously, I suppose.