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February 4

sum of cubes

Hi all - I hope this question hasn't been asked here before, and thanks in advance for any help...

I was fooling around with some calculations, and I suddenly noticed that the sum of the cubes of any two natural numbers seems to always be exactly divisible by the sum of those numbers - that is, if a and b are natural, then (a^3 + b^3)/(a+b) is also a natural number. Is this a "real" law, or have I simply hit upon some rare combinations by chance that it seems to work for? If it is a real law, does it have a name, and is there a simple proof for it? Thanks, Grutness...wha? 11:12, 4 February 2019 (UTC)[reply]

actually, I may have gained an extra insight: it looks like (a^3 + b^3) = (a^2 + b^2 - ab)*(a+b). Grutness...wha? 11:18, 4 February 2019 (UTC)[reply]

Yes, that is exactly right. If you expand out

(a+b)(a^{2}-ab+b^{2})

you get

a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2})=a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}=a^{3}+b^{3}

. So

a^{3}+b^{3}

is divisible by

a+b

and the ratio is

a^{2}-ab+b^{2}

, an integer.

It is also true that

a^{5}+b^{5}=(a+b)(a^{4}-a^{3}b+a^{2}b^{2}-ab^{3}+b^{4})

so it is divisible by

a+b

, and in general for every odd positive integer n, you have

a^{n}+b^{n}=(a+b)\sum _{i=0}^{n-1}(-1)^{i}a^{n-1-i}b^{i}

. -- Meni Rosenfeld (talk) 12:54, 4 February 2019 (UTC)[reply]

(ec) The general form is

a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+\dots +ab^{n-2}+b^{n-1})

, a typical factorisation of a binomial using a telescoping sum. Your example is the special case

a^{3}-(-b)^{3}=(a-(-b))(a^{2}+a(-b)+(-b)^{2})

. —Kusma (t·c) 13:00, 4 February 2019 (UTC)[reply]

Meni Rosenfeld you may want to fix your math, you've got your final piece of the first equality being

a^{2}+b^{3}

Naraht (talk) 15:11, 4 February 2019 (UTC)[reply]

Fixed, thanks. -- Meni Rosenfeld (talk) 18:30, 4 February 2019 (UTC)[reply]

We have the article sub-section Factorization#Recognizable patterns. Loraof (talk) 18:51, 4 February 2019 (UTC)[reply]

Another way to see this is by considering

a^{3}+b^{3}

to be a polynomial function of the variable

b

. Then because this polynomial has a zero at

b=-a

, it contains a factor of

\left(b+a\right)

. Count Iblis (talk) 21:48, 4 February 2019 (UTC)[reply]

Many thanks everyone. Grutness...wha? 00:01, 5 February 2019 (UTC)[reply]

Are there any models of the Euclidean plane that use non-Euclidean geometry?

I am interested in non-Euclidean geometry. I've read that there are several models of hyperbolic geometry that translate it into something that can be depicted in Euclidean space. (I haven't heard that much about models of elliptic geometry for some reason...) Are there any models that translate Euclidean geometry into something that can be depicted in non-Euclidean space (such as hyperbolic or elliptic)? I suspect that people haven't done much research on that, as we (probably) live in an Euclidean world, but I'm asking just in case there are any. I have done a Google search, but I only found models of non-Euclidean geometry, not Euclidean geometry. Diamond Blizzard talk 17:46, 4 February 2019 (UTC)[reply]

You could define a coordinate system in hyperbolic space similar to the way you define a coordinate system in euclidean space. Specifically, pick two lines at right angles to call the x- and y-axes and define the coordinates of a point P to be (x_P, y_P) where x_P is the signed distance from P to the y-axis and y_P is the signed distance from P to the x-axis. This identifies R² with the hyperbolic plane and we already know how to identify R² with the Euclidean plane. In most models of the hyperbolic plane in Euclidean space the hyperbolic lines don't correspond with Euclidean lines but with curves, usually circles. A similar thing happens here; the Euclidean lines defined by ax+by=c don't correspond to hyperbolic lines but to curves. For the lines x=c and y=c the curve is called a hypercycle. I don't know about the more general form ax+by=c, but I suspect it's more complex. You might also say that Euclidean space is what you get in the limit when you zoom in on hyperbolic space, though I don't know if that idea can be made rigorous.

As to whether we live in an Euclidean world, that's really a matter for physics but General Relativity says no. Nor is it a uniform hyperbolic space but one where mass affects curvature. On cosmic scales the universe is apparently very flat, i.e. Euclidean-like, but we can only observe part of the universe so who knows? See Shape of the universe for details. --RDBury (talk) 03:04, 5 February 2019 (UTC)[reply]

{reply to|RDBury}} Interesting. Are there any specific sources you used to give this answer? I would like to know which sources give information on this topic. Also, as for the universe being Euclidean, I have read the Shape of the universe article. It is just that the universe seems very close to flat on large scales, and I haven't heard of any particular reason that it would be different outside the observable universe. Diamond Blizzard talk 05:17, 5 February 2019 (UTC)[reply]

Apparently Diamond Blizzard tried to ping RDBury above, but something went wrong. So I make a double ping to notify both of them here. :) CiaPan (talk) 07:56, 5 February 2019 (UTC)[reply]

No specific sources (my Google search didn't turn up anything either) but I think the idea is valid; I'm sure someone will point it out if not. It's not really the kind of thing that could be called a theorem so I'm not sure why it would be included in a textbook or research paper. Really all you need for the idea to work is a homeomorphism between the underlying topologies, so there are all sorts of possibilities: a model of the Euclidean plane in the upper half plane, or the interior of a circle, or a sphere with a point removed. Or you could model spherical geometry by adding a point to the Euclidean or hyperbolic plane. The projective plane could be made to work as well if you perform a little surgery; for example you easily get the Euclidean plane by removing a line from the projective plane, but you can produce the projective plane by identifying antipodal points on a sphere. Some of these are well known because they are simple and useful for calculations, e.g. the Riemann sphere, but most aren't. --RDBury (talk) 14:49, 5 February 2019 (UTC)[reply]

@RDBury: Thanks! Just one thing I remembered: you brought up mass affecting the curvature of the universe, and there's the fact that while the universe seems to be flat, there is always a margin of error in measurements. Does this mean that modern physics can, in fact, state what gravity would be like in hyperbolic space? I'm interested in this because the author, in this link http://www.stevejtrettel.com/living-in-hyperbolic-space.html stated that he does not know how gravity would actually work in hyperbolic space. There's also the fact that you don't really have parallel lines in hyperbolic space, so it gets interesting, as mentioned in this link: http://zenorogue.blogspot.com/ (although, admittedly, it is in the context of game design, for a game that explicitly does not take place in a relativistic world - but it's still something that I think about). Hope this isn't off-topic; I just remembered it because you mentioned relativity and mass in this context. Diamond Blizzard talk 03:50, 6 February 2019 (UTC)[reply]

Well, this is getting more into physics than math, and somewhat speculative physics at that, but my understanding is that in general relativity you really can't separate gravity from the curvature of space, in particular the local curvature of space-time caused by matter. In Euclidean, hyperbolic and spherical geometry the curvature of space is constant, so the paths of two objects would be independent of each other; effectively gravity would not exist. If you want to ignore local curvature of space then you're talking Newtonian gravity which wouldn't be called 'modern' physics, but inverse square laws probably would not apply in non-Euclidean space. As to what would replace inverse square laws, I'm pretty sure the link you gave would answer that better than I could. On the Rogue link, there are several possible definitions of parallel, which are equivalent in Euclidean geometry but not in other geometries, so whether there are parallel lines in hyperbolic geometry depends on what definition you use. Certainly there are non-intersecting lines in hyperbolic geometry, but no such thing as lines that have constant distance from each other. --RDBury (talk) 13:33, 6 February 2019 (UTC)[reply]

You might like these two Math Stack Exchange questions. In particular, just as the surface of a sphere provides a model of spherical geometry in Euclidean space, so the surface of a horosphere provides a model of Euclidean geometry in hyperbolic space. But you cannot embed R^m into Sⁿ for any m and n, so creatures who live in spherical geometry will require a less obvious model, as we do for hyperbolic geometry. Double sharp (talk) 14:45, 6 February 2019 (UTC)[reply]