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.

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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?

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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?

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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).

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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.]

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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 260^th 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 260^th terms there. So it’s table time again!

The 260^th 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

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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

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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.)

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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