Talk:Squared triangular number

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)


 * Nicomachus observes it but doesn't offer a proof. I haven't looked up Aryabhata's discussion, but the general style of the early Indian mathematicians was to state results without proof.  In fact this was also common in European algebra down to the 17th century.  For example, Descartes stated his 'rule of signs' without anything resembling a proof.  I think this was partly because mathematicians often regarded their methods as 'trade secrets', which they could use to solve challenge problems, but also because until the principle of mathematical induction was understood, a lot of problems were difficult to prove rigorously with the available methods.2A00:23C8:7907:4B01:B180:FF9C:33B1:1040 (talk) 21:29, 9 November 2021 (UTC)Added by same commenter:  the passage in Nicomachus is in Chapter 20 of Book 2 of his 'Arithmetic', not in Chapter 20 of the whole work, which would be in Book 1.2A00:23C8:7907:4B01:5064:8DD0:F21C:FD4B (talk) 15:38, 17 November 2021 (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. pm a  00:22, 19 November 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 the RHS is obviously just the Cartesian product of two isosceles right triangles 0≤x≤y≤N and 0≤z≤w≤N in the xy and zw planes, a natural geometric interpretation of a squared triangular number. —David Eppstein (talk) 07:01, 16 February 2021 (UTC)
 * Thank you for this explanation; it's a lot clearer than the paragraph in the article itself. Perhaps you'd like to improve that paragraph - or at least give it a tag that you feel is more appropriate than "Confusing"? PatricKiwi (talk) 07:28, 16 February 2021 (UTC)

Triangular numbers of squares
I recently added that these numbers are the alternating sums of Triangular numbers of squares, similar to how triangular numbers are the alternating sums of squares, however this was removed because of a lack of proof: 0+1 is 1 1+9 is 10 ( triangle 4) 9+36 is 45 (triangle 9) 36+100 is 136 (triangle 16) 100+225 is 325 (triangle 25) 225+441 is 666 (triangle 36) 441+784 is 1225 (triangle 49) 784+1296 is 2080 (triangle 64) 1296+2025 is 3321 (triangle 81) 2025+3025 is 5050 (triangle 100) KlokkoVanDenBerg (talk) 09:29, 7 April 2022 (UTC)
 * It was not removed for lack of proof. It was removed for lack of a published source. Also, 10, 45, 136, etc., are not squared triangular numbers. —David Eppstein (talk) 16:14, 7 April 2022 (UTC)
 * They are triangular numbers of squares KlokkoVanDenBerg (talk) 17:38, 7 April 2022 (UTC)
 * Like how 10 is the 4th triangular number, 45 is the 9th triangular number, 136 is the 16th, and so on. KlokkoVanDenBerg (talk) 17:40, 7 April 2022 (UTC)
 * Triangular numbers of squares are a different topic than this article. This article is about squares of triangular numbers. —David Eppstein (talk) 17:40, 7 April 2022 (UTC)
 * I know, but the sum of two consecutive Squared triangular numbers is a Triangular numbers of squares KlokkoVanDenBerg (talk) 17:42, 7 April 2022 (UTC)