Talk:Pentagonal number

Generalized

I don't know what is the point of generalized pentagonal numbers, as opposed to regular pentagonal numbers, so if someone knows, it would be appreciated if they added it to the article. PrimeFan 21:17, 4 Mar 2004 (UTC)

There's a point to the regular ones? The generalized ones are the scrunty little 'pentagons' inside the patterns for the 'regular' ones. Personally, I can't see why a ten-dot pentagon with two dots in the middle is any more regular than the two dots on their own. Grant 18:37, 25 November 2006 (UTC)

7719539042 2409:4051:387:3722:0:0:20A6:90A1 (talk) 03:12, 24 January 2024 (UTC)

Represents a pentagon

The article as it appears in 2010 does not (no lnger?) seems to have the phrase "represents a pentagon." The numbers in fact do not represent pentagons; rather pentagons (or superpositions of pentagons) are one way of represnrting the numbers. And not the only way. For an alternative representaiton, which treats generalized pentagonal numbers equally rtegardless of whether they are pentagonal numbers proper, see the section of the article on centered hexagonal numbers. 170.170.59.138 (talk) 23:39, 15 July 2010 (UTC)OL3

The article should explain what "represents a pentagon" means. 84.139.0.238 17:38, 7 February 2006 (UTC)

You're absolutely right. This is one of those things that need to be explained precisely because they seem so obvious.

I looked to Mathworld for guidance. Their article on the topic has a graphic of dots arranged in pentagons, but nothing in words. So it's a challenge for us to come up with appropriate words. PrimeFan 21:16, 7 February 2006 (UTC)

You're having a laugh. It's not remotely obvious. Give that one of the chief features of a regular pentagon is its rotational symmetry, might one not expect a pattern of dots that represents such a figure to have its own rotational symmetry? In fact there is no sense in which the number 12, say, represents a pentagon. Grant 17:33, 25 November 2006 (UTC)

I created a gif image representing the first six pentagonal numbers in pentagon form, hopefully it clears things up. timrem 02:21, 15 March 2007 (UTC)

Music to my ears

One can see 1, 5, 12, 22, 35... as pentagonal patterns. One can also see this pattern in human music. 1 from monochords, 5 for 'pentatonic scales, 12 for Western dodecaphonic (12 tone) system in music, and 22 for the 'Sruti' (22-tone) system in Indian music. Some jazz musicians have experimented with 35-tone music. Figurate numbers should find a place along side Pythagoras' "music of the spheres" in the bridging of Math and Music. If you like, we can discuss this more over at my forum: http://forums.delphiforums.com/figurate/start —Preceding unsigned comment added by 66.236.228.11 (talk • contribs)

Comparison to triangular numbers dubious

Currently the lede states that unlike triangular numbers, "the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical". Well, if you construct the triangular numbers as right-angled triangles, they are not rotationally symmetrical either. Wouldn't it be more correct to express the difference by stating that the constructed points cannot all be a subset of a regular grid? References of expert attention would be good. --Cedders^tk 09:19, 5 November 2018 (UTC)

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Is generalized pentagonal numbers * n + 1 can be primes for all positive integers n other than 24 and 25? Also, if there a formula of the sum of reciprocals for (generalized pentagonal numbers * n + 1) for positive integers n?

Generalized pentagonal numbers * n + 1 can be primes, for all positive integers n, except 24 and 25, since generalized pentagonal numbers * 24 + 1 are squares, and generalized pentagonal numbers * 25 + 1 are also generalized pentagonal numbers, this is like that triangular numbers * n + 1 (centered n-gonal numbers) can be primes, for all positive integers n, except 8 and 9, since triangular numbers * 8 + 1 are squares, and triangular numbers * 9 + 1 are also triangular numbers.

In base 25, all repunits are generalized pentagonal numbers, and in base 9, all repunits are triangular numbers, and since no generalized pentagonal numbers > 7 are primes, and no triangular numbers > 3 are primes, in these two bases there are no generalized repunit primes, also, there is a formula of the sum of reciprocals for the centered n-gonal numbers, i.e. the sum of reciprocals for (triangular numbers * n + 1) for positive integers n, but is there a formula of the sum of reciprocals for (generalized pentagonal numbers * n + 1) for positive integers n? ——220.132.54.182 (talk) 09:00, 29 December 2021 (UTC) — Preceding unsigned comment added by 220.135.64.212 (talk)

Tests for pentagonal numbers unclear

The description says how to test whether a number x is a non-generalized pentagonal number given an equation. Then it says x is a non-generalized pentagonal number iff n is a natural number.

Then it gives a perfect square test to see if x is a generalized pentagonal number and says this is sufficient to tell if x is a gen. pent. number. But then it confusingly says "For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if [equation]."

It said the perfect square test was for the generalized pent nums, not non-generalized. If there is another condition required to check if a number x is a generalized pentagonal number, then the earlier statement that the first check is sufficient, is false, and if the additional test is needed for non-generalized pent num, then the earlier statement that x is a non-generalized pentagonal number if and only if n is a natural, is false, because it would also need this 2nd condition checked.

So which is it? Stickmcskunky (talk) 04:29, 24 November 2024 (UTC)