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.
I vote this the best proof. All you need to know to understand it is to know how multiplication, addition and exponentiation works. You could probably show this to a child in a sixth grade or so, and have them understand it. This is really good!
>> Apparently I'm good enough at math to do the proofs, but not to write the exercises.
Took a look on it, seems like a highly particular / specialized area of mathematics. It's like computer science, can't know them all. If you work all day with some area, say compilers or databases or financial software or what else, you'd be a whizz at it while it's unreasonable to expect someone from a different domain be able of more than a superficial understanding of what you write.
I'm pretty good at math but like with computers, I don't have the compulsion to dive deep into an unfamiliar domain just for the sake of it. So commenting on the article: cool, now I know how these problems are formed and in the very unlikely domain I'll need to produce one, I know where to look. Likely this will never happen, though.
> seems like a highly particular / specialized area of mathematics. It's like computer science, can't know them all
As a non-mathy, I'm interested in whether the idea that being good/able to provide proofs in one area, automagically makes one proficient in another is customary in the field or rejected quite early on when choosing a math specialisation?
It is not typical that being good in one area of mathematics makes one proficient in another — mathematics has a lot of depth, and to reach the frontier in any specialization requires years of study. However:
- There is a story/legend told about Erdős, where he was so good at problem-solving/proofs that he once solved a problem in another area after asking for the definitions of the terms in the problem. (The fact that this story is told illustrates that it is not commonplace.)
Well there is common stuff that one must be familiar with to be proficient and marginal stuff that you can get by without knowing because chances is you're not going to need it.
Like take for instance financial mathematics where I had some special interest, it's totally oblivious to areas such as geometry or number theory. I never had to figure out if a polynomial is divisible by 6 for instance :)
Like computer science, there's the common algorithms stuff but being an expert in web development doesn't help you much in writing high frequency trading server code, and the other way around.
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.
As the post mentions in passing, the integer-valued polynomials are completely characterized by the property that when written as a sum of {c_i (x choose i)}, all the coefficients c_i are integers. I imagine this is where most of the exercises actually come from. For example, using [3 1 4 1 5 9], the polynomial {3 + 1·x + 4·x(x-1)/2 + 1·x(x-1)(x-2)/6 + 5·x(x-1)(x-2)(x-3)/24 + 9·x(x-1)(x-2)(x-3)(x-4)/120} simplifies to 1/120 (9x^5 - 65x^4 + 185x^3 + 5x^2 - 14x + 360), so you could use it to generate exercises like:
- Prove that 9x^5 - 65x^4 + 185x^3 + 5x^2 - 14x + 360 is always a multiple of 120
n = 0 or 1 modulo 2. So we have to check out only two cases module 2, and these cases trivial. To prove the problem note that 6 = 2 * 3 and then is trivial to see that the polynomial is=0 modulo 2 if n=0,1 modulo 6and then check it is =0 modulo 3 for n=0,1,2 modulo 3 and you are done.
That article is mostly about speed. The following seems like the one thing that might be relevant:
> Naively, you could take the random integer and compute the remainder of the division by the size of the interval. It works because the remainder of the division by D is always smaller than D. Yet it introduces a statistical bias
That's all it says. Is the point here just that 2^31 % 17 is not zero, so 1,2,3 are potentially happening slightly more than 15,16? If so, this is not terribly important
It is not uniformly random, which is the whole point.
> That article is mostly about speed
The article is about how to actually achieve uniform random at high speed. Just doing mod is faster but does not satisfy the uniform random requirement.
additional to the other excellent comments they will become non-uniform once you start deleting records. that will break all hopes you might have had in modulo and percentages being reliable partitions because the "holes" in your ID space could be maximally bad for whatever usecase you thought up.
You don't ban people from consuming sugar, you ban companies from mass-producing food with unhealthy amounts of it. It's about friction and incentives. If you want to buy a bag of sugar and bake a cake that's fine.
shades of grey and tradeoffs. gambling has a very clear and well studied negative effect on a small but sizeable percent of people and its not that great for everyone else. its my opinion were better off without it
According to whom? I’ve developed and run Wordpress sites for nearly 15 years and never enabled revisions on a single one.
This whole line of thinking—one person defining the “core tenet” or whatever—seems directly contrary to the ethos of open source. If Matt doesn’t think people should be able to turn off revisions, he should put that in the license. Otherwise, users can do what they want and open source leaders should celebrate that.
Idk much about him but stacking school busses on top of each other with a crane or driving a train into a sinkhole seem like pretty interesting things to do. Better than geeking out over the bloodiest Mortal Kombat fatality or whatever I was doing at that age. What's an example of the more "crap" content?
They appear experienced at navigating this confusion