Talk:Squared triangular number
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Visual proof
It would be nice to include the picture proof from http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32, since it is clearer than the one already on this page, but I am not sure about the copyright status of the images. Please advise.
Loeb 02:17, 30 October 2007 (UTC)
- It is a pretty proof. I think only the second of the two images there can easily stand on its own. BUT, I would like to know the source of the proof. Has it been published anywhere? Is it the same as one of the existing proofs already known in the article? I don't think a web page makes an adequate reference for a problem with as much history as this one. —David Eppstein 02:22, 30 October 2007 (UTC)
Proof
Who was the first to prove this theorem? I've seen it called Nicomachus' Theorem, but did he prove it or merely observe it? If not, did Āryabhaṭa? Al-Karajī?
CRGreathouse (t | c) 18:22, 31 August 2009 (UTC)
Sum of the first "n" cubes - even cubes - odd cubes (geometrical proofs)
Starting from the basic idea described in the first animation, we introduced a new procedure to obtain formulas for summing the first "n" cubes, even cubes and odd cubes. This method, called "Successive Transformations Method", consists in an inductive handling of a geometric model, in order to obtain another equivalent which gives evidence of the searching formulas.
See the animations.
Consider the final transformation that you see in the second animation, and we compare this figure with the square base parallelepiped that contains it. We expect that the ratio between these figures becomes 1/2 to infinity. Performing calculations with Excel, you see that this is true. In addition you encounter these other amazing results:
n
lim (Σn n3)/(Σn n).n2 = 1/2
n→∞ 1
n
lim (Σn n5)/(Σn n).n4 = 1/3
n→∞ 1
n
lim (Σn n7)/(Σn n).n6 = 1/4
n→∞ 1
n
lim (Σn n9)/(Σn n).n8 = 1/5
n→∞ 1
which, by induction, can be generalized in a formula. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers. The induction principle enshrines the validity of this "theorem". --79.17.53.156 (talk) 09:21, 3 August 2013 (UTC)
- Very interesting, but we're only supposed to put things into the article that are supported by published sources, see WP:V and WP:RS. People's own thoughts have no place in the encyclopaedia no matter how good they are, see WP:OR. Dmcq (talk) 20:10, 7 October 2013 (UTC)
- This my work is currently being published on italian magazine. — Preceding unsigned comment added by Ancora Luciano (talk • contribs) 08:39, 9 October 2013 (UTC)
- Honestly, the evaluation of these limits is a standard exercise of calculus. No more reasons to be included in any encyclopaedia, than the evaluation of 129+934. It's a computation. Just apply the definition of Riemann integral for xk. pma 00:22, 19 November 2013 (UTC)
- This my work is currently being published on italian magazine. — Preceding unsigned comment added by Ancora Luciano (talk • contribs) 08:39, 9 October 2013 (UTC)
Probabilistic Interpretation?
The LHS indeed is correct, but the RHS doesn't make any sense. — Preceding unsigned comment added by 1.162.222.80 (talk) 07:39, 16 April 2016 (UTC)
- Indeed, both probabilities coincide with the left and right side of the Nicomachus identity, but the events are certainly not equivalent. In fact, one can only conclude their probabilities are equal because of the Nicomachus identity. There are no further references either, and I could not find anything else in google. I think this is an original idea someone posted. — Preceding unsigned comment added by 61.127.95.156 (talk) 13:08, 12 February 2018 (UTC)
- I agree; that section is at best confusing, and at worst wrong. I've added a "Confusing section" tag. PatricKiwi (talk) 06:54, 16 February 2021 (UTC)
- It needs a source, but there is a very natural four-dimensional polyhedral interpretation in which this equality of probabilities is analogous to the equality of polyhedral numbers. The LHS is the volume of the subset max(x,y,z)≤w of the unit 4-cube, which by Cavalieri's Principle in the w-direction is just the integral over all choices of w of the cube max(x,y,z)≤w in the 3-dimensional cross-section for that w, and the RHS is the volume of a Cartesian product of two isosceles right triangles 0≤x≤y≤1 and 0≤z≤w≤1 in the xy and zw planes. So it's the same principle, a sum (or this time integral) over cubes equals the square of a triangle. To me that's so similar as to make it obviously not coincidental. —David Eppstein (talk) 07:01, 16 February 2021 (UTC)