Polygonal Numbers

A polygonal number is defined as “A type of figurate number which is a generalization of triangular, square, etc., numbers to an arbitrary n-gonal number. The above diagrams graphically illustrate the process by which the polygonal numbers are built up.” (Mathworld.wolfram.com) Every student of school mathematics knows about the square numbers, and many know about the triangular numbers as well. But less familiar are the pentagonal, hexagonal, etc. varieties.

Even less well known is the fact that each of those types of numbers has a cousin of sorts, called the centered polygonal numbers. Yes, the regular triangular numbers have their corresponding “centered” form. The same is true for the squares, pentagonals, hexagonals, etc. (See diagram below.)

Therefore, our definition for these numbers is “A figurate number in which layers of polygons are drawn centered about a point instead of with the point at a vertex.” (Mathworld.wolfram.com)

Many facts and theorems are known about polygonal numbers, especially of the squares and triangulars. We wouldn’t be able to talk about the Pythagorean Theorem if it weren’t for the squares, just to mention the most famous example of all. And the triangulars arise whenever we are concerned with the sum of consecutive integers, from 1 to n.

What I want to do in this page of WTM is present some ideas that are not normally covered in an average school math class, yet ideas that are well within the understanding of most students. First, we will show the algebraic formulas for both the regular and centered polygonal numbers, up to a level seldom discussed: a 30-sided polygon!

The Formulas
Number
of Sides
Regular
form
Centered
form
3 n(n + 1)/2 (3n2 – 3n + 2)/2
4 n2 2n2 – 2n + 1
5 n(3n – 1)/2 (5n2 – 5n + 2)/2
6 n(2n – 1) 3n2 – 3n + 1
7 n(5n – 3)/2 (7n2 – 7n + 2)/2
8 n(3n – 2) 4n2 – 4n + 1
9 n(7n – 5)/2 (9n2 – 9n + 2)/2
10 n(4n – 3) 5n2 – 5n + 1
11 n(9n – 7)/2 (11n2 – 11n + 2)/2
12 n(5n – 4) 6n2 – 6n + 1
13 n(11n – 9)/2 (13n2 – 13n + 2)/2
14 n(6n – 5) 7n2 – 7n + 1
15 n(13n – 11)/2 (15n2 – 15n + 2)/2
16 n(7n – 6) 8n2 – 8n + 1
17 n(15n – 13)/2 (17n2 – 17n + 2)/2
18 n(8n – 7) 9n2 – 9n + 1
19 n(17n – 15)/2 (19n2 – 19n + 2)/2
20 n(9n – 8) 10n2 – 10n + 1
21 n(19n – 17)/2 (21n2 – 21n + 2)/2
22 n(10n – 9) 11n2 – 11n + 1
23 n(21n – 19)/2 (23n2 – 23n + 2)/2
24 n(11n – 10) 12n2 – 12n + 1
25 n(23n – 21)/2 (25n2 – 25n + 2)/2
26 n(12n – 11) 13n2 – 13n + 1
27 n(25n – 23)/2 (27n2 – 27n + 2)/2
28 n(13n – 12)/2 14n2 – 14n + 1
29 n(27n – 25)/2 (29n2 – 29n + 2)/2
30 n(14n – 13) 15n2 – 15n + 1

Hey! Do you see a pattern in the table? If you do, perhaps you could write a general formula for it; then you could give the formula for any n-gonal number of either type, without using the table, and even beyond 30 sides.


And now for some numbers…

Our next chart will give us some actual numbers for the polygons up to decagons.

The First Nine Terms
Name Regular Centered
triangular 1, 3, 6, 10, 15, 21, 28, 36, 45, … 1, 4, 10, 19, 31, 46, 64, 85, 109, …
square 1, 4, 9, 16, 25, 36, 49, 64, 81, … 1, 5, 13, 25, 41, 61, 85, 113, 145, …
pentagonal 1, 5, 12, 22, 35, 51, 70, 92, 117, … 1, 6, 16, 31, 51, 76, 106, 141, 181, …
hexagonal 1, 6, 15, 28, 45, 66, 91, 120, 153, … 1, 7, 19, 37, 61, 91, 127, 169, 217, …
heptagonal 1, 7, 18, 34, 55, 81, 112, 148, 189 1, 8, 22, 43, 71, 106, 148, 197, 253, …
octagonal 1, 8, 21, 40, 65, 96, 133, 176, 225, … 1, 9, 25, 49, 81, 121, 169, 225, 289, …
nonagonal 1, 9, 24, 46, 75, 111, 154, 204, 261, … 1, 10, 28, 55, 91, 136, 190, 253, 325, …
decagonal 1, 10, 27, 52, 85, 126, 175, 232, 297, … 1, 11, 31, 60, 101, 151, 211, 281, 361, …

Now that we have some numbers, what should we do with them? If I may paraphrase an old popular song by Nancy Sinatra, these numbers are made for adding! So consider this…

We again turn to Mathworld for some vital information: Fermat’s Polygonal Number Theorem
In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. Gauss proved the triangular case, and noted the event in his diary on July 10, 1796, with the notation

What that little cryptic notation means is that all whole numbers can be expressed as the sum of three, or fewer, triangular numbers. Here is an interesting example:

100 = 91 + 6 + 3 = T13 + T3 + T2

100 = 55 + 45 = T10 + T9

This illustrates that sometimes a number has two possibilities, with 3 or 2 terms. Nice, huh?


Turning now to the case of the squares… Fermat said that all whole numbers can be expressed as the sum of four, or fewer, square numbers. Let’s look at this example:

50 = 49 + 1 = S7 + S1

50 = 25 + 25 = S5 + S5

50 = 25 + 16 + 9 = S5 + S4 + S3

50 = 36 + 9 + 4 + 1 = S6 + S3 + S2 + S1

Notice that there were expressions with 2, 3, and 4 terms. Thereby, arises an interesting idea: given a particular number, how many different expressions can be found? I challenge you to research this and report back to me. Ok?


One more time… For the case of the pentagonals, we can use up to five of them to express all whole numbers. Let’s check out the situation for the number 2002.

2002 = 1001 + 1001 = P26 + P26

2002 = 1520 + 477 + 5 = P32 + P18 + P2

2002 = 1820 + 176 + 5 + 1 = P35 + P11 + P2 + P1

2002 = 1717 + 176 + 92 + 12 + 5 = P34 + P11 + P8 + P3 + P2

As before, we have demonstrated that we can achieve our goal with 2-5 terms. In fact, there are many more such ways to do it than presented here. Finding all possible ways is now more difficult, (unless one uses a computer program).


The Other Side of the Story

So far we have only been working with the regular polygonal numbers. Let’s now look at the centered case. The natural question to ask should be: does there exist an analogue to Fermat’s theorem, as discussed above? Specifically, are three CTN’s (Centered Triangular Numbers) sufficient to express all whole numbers?

The best way to answer that is to start small and work your way up. Here is a chart for the numbers from 1 to 10. Recall, the set of CTN’s is {1, 4, 10, 19, …}.

The CTN Analogue
No. expression No. expression
1 1 6 4 + 1 + 1
2 1 + 1 7 none
3 1 + 1 + 1 8 4 + 4
4 4 9 4 + 4 + 1
5 4 + 1 10 10

Well, I guess that about answers our question, doesn’t it? And it didn’t take long either.

However, it brings to mind yet another question — what is the next impossible number?

And the next? And the next?

Then what happens when this idea is extended to CSN’s (Centered Square Numbers) and CPN’s (Centered Pentagonal Numbers)? What are the impossible values when using these other sets of numbers? And beyond?

[Remember: you can use up to 4 CSN's and 5 CPN's, and so on, in the expressons.]


Special Numbers

Another popular activity when one is faced with a long list of numbers is to search for the presence of numbers with special characteristics, such as squares, cubes, or palindromes. Let’s first consider the modern favorite of many mathematicians: palindromes.

The “mother of all websites” dealing with palindromes undoubtedly is World!Of Numbers, edited by Patrick De Geest. In his site you can find an extensive treatment of palindromes that are also triangular numbers, and squares as well. In fact, he gives data for the pentagonals up to the nonagonals. We heartily encourage you to visit that site; you will be justly rewarded for your time and efforts.

However, all the data to be found there uses only the regular type of polygonals; there is nothing mentioned about the centered type. Here is our attempt to fill in that gap of trivia information. (Note: At present, our search only shows results up to n = 300 and for k-gonals from k = 3 to 40. We also omit any single-digit palindromes as being trivial in this context.)

The Palindromes
k n CPkN(n) Prime Factorization
3 101 15151 109 x 139
174 45154 2 x 107 x 211
211 66466 2 x 167 x 199
249 92629 211 x 439
257 98689 prime
4 10 181 prime
13 313 prime
17 545 5 x 109
5 8 141 3 x 47
9 181 prime
65 10401 3 x 3467
6 18 919 prime
7 3 22 2 x 11
20 1331 113
8 6 121 112
51 10201 1012
56 12321 32 x 372
61 14641 114
9 4 55 5 x 11
12 595 5 x 7 x 17
10 2 11 prime
5 101 prime
11 5 111 3 x 37
7 232 23 x 29
11 606 2 x 3 x 101
12 727 prime
62 20802 2 x 3 x 3467
12 5 121 112
6 181 prime
16 1441 11 x 131
46 12421 prime
13 5 131 prime
14 5 141 3 x 47
8 393 3 x 131
9 505 5 x 101
15 5 151 prime
10 676 22 x 132
52 19891 prime
16 5 161 7 x 23
46 16561 prime
17 5 171 32 x 19
93 72727 prime
18 3 55 5 x 11
5 181 prime
8 505 5 x 101
40 14041 19 x 739
19 5 191 prime
20 4 121 112
21 2 22 2 x 11
9 757 prime
36 13231 101 x 131
120 149941 11 x 43 x 317
255 680086 2 x 11 x 19 x 1627
22 none in this range yet
23 7 484 22 x 112
29 9339 3 x 11 x 283
24 7 505 5 x 101
25 4 151 prime
36 15751 19 x 829
289 1040401 101 x 10301
26 30 11311 prime
27 8 757 prime
28 35 16661 prime
29 3 88 23 x 11
30 4 181 prime
94 131131 7 x 11 x 13 x 131
260 1010101 73 x 101 x 137
31 69 72727 prime
32 2 33 3 x 11
10 1441 11 x 131
26 10401 19 x 739
251 1004001 3 x 334667
33 260 1111111 239 x 4649
34 25 10201 1012
43 30703 prime
172 500005 5 x 11 x 9091
35 25 10501 prime
28 13231 101 x 131
33 18481 prime
36 25 10801 7 x 1543
260 1212121 prime
37 none in this range yet
38 31 17671 41 x 431
39 52 51715 5 x 10343
260 1313131 17 x 77243
40 3 121 112
9 1441 11 x 131
27 14041 19 x 739
225 1008001 prime

While studying the results above, I saw two rather interesting numbers: 1212121 and 1313131. Not only do they share an obvious digital pattern of 1d1d1d1, but they both are the 260th term in their respective orders (k = 36 and 39).

Now if we look back at the 33-gonal and 30-gonal lists, we see 1111111 and 1010101. As one might begin to suspect by now, they are the 260th terms there. So it’s table time again!

The 260th Term Case
k Number Prime Factorization
30 1010101 73 x 101 x 137
33 1111111 239 x4649
36 1212121 prime
39 1313131 17 x 77243
42 1414141 43 x 32887
45 1515151 11 x 181 x 761
48 1616161 prime
51 1717171 199 x 8629
54 1818181 31 x 89 x 659
57 1919191 29 x 66179

Stop the Presses!!! (4/22/2002)

Just in to the editorial offices of WTM! Patrick De Geest has sent in a pair of 6-packs…of palindromes for the missing data in the chart of Palindromes above. Here it is.

Palindromes for k = 22 & 37
k n CPkN(n) Prime Factorization
22 4156 189949981 13 x 14611537
524962 3031430341303 7 x 13 x 33312421333
321895111 1139781083801879311 13 x 53 x 163 x 15259 x 665102447
358542860 1414082803082804141 7 x 19 x 67 x 349 x 2671 x 170235089
362349816 1444271276721724441 83 x 17400858755683427
422820435 1966548318138456691 17 x 191 x 5346613 x 113277481
37 378 2636362 2 x 163 x 8087
2400 106515601 43 x 2477107
407157 3066863686603 prime
2835585 148749979947841 859 x 56099 x 3086801
3443283 219339595933912 23 x 27417449491739
6792834 853637858736358 2 x 17 x 25106995845187

Also, my friend and colleague from Romania, Andrei Lazanu, sent along some important data regarding the matter above about the impossible cases for numbers to be expressed as the sum of 3, or fewer, CTN’s. According to the program he wrote, there are over 70 numbers less than 200 that can not be decomposed in this way. How many can you find? Can you find all of them?

Andrei also was kind enough to provide WTM with some information about how many ways 2002 can be expressed with 5, or fewer, regular pentagonal numbers. It can be done in 166 ways. [See examples above.]

In addition, he informs us that the same task can be achieved in 31 ways with regular triangular numbers, and in 101 ways using regular square numbers. (Thanks, Andrei.)


Update (5/13/02)

De Geest has provided WTM with some more CTN palindromes. Here they are (as of 4/25/02).

More CTN Palindromes
(n < 3,711,895,911)
n CTN Prime Factorization
920 1690961 29 * 58309
1258 3162613 101 * 173 * 181
1263 3187813 prime
1622 5258525 52 * 17 * 12373
1707 5824285 5 * 17 * 68521
170707 58281418285 5 * 39821 * 292717
904281 1635446445361 109 * 9461 * 1585889
1258183 3166046406613 173 * 13537 * 1351913
7901015 124852060258421 21589 * 5783133089
8659930 149988757889941 17 * 852013 * 10355321
12458598 310433303334013 101 * 3073597062713
17070707 582818040818285 5 * 89 * 461693 * 2836741
80472265 12951570707515921 13 * 17 * 113 * 1609 * 322326253
1616689804 5227371841481737225 52 * 941 * 104917 * 2117911937
1680689789 5649436330336349465 5 * 29 * 761 * 51197936746897
1705387644 5816694029204966185 5 * 1163338805840993237

Next (5/13/02), Andrei has extended the matter of Centered Hexagonal Numbers (CP6N) if ever so slightly…

More CP6N Palindromes
(n < 1000)
n CP6N Prime Factorization
601 1081801 7 * 154543
630 1188811 13 * 19 * 4813
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One thought on “Polygonal Numbers

  1. The list of indices of palindromic centered triangular numbers differs from the one
    in the OEIS, http://oeis.org/A005448 . 3*n(n-1)/2+1 is palindromic for n = 1, 2, 101, 174, 211, 249, 257, 1822, 2070, 20795,… The update of 5/13/02 does not
    fit in there.
    Richard

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