Figurate Numbers
by
George Jelliss
June 2008
with additions
November 2008
Visualisation of Numbers
The visual representation of the number of elements in a set by an array of small
counters or other standard tally marks is still seen in the symbols on dominoes or playing cards,
and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate.
Bear with me while we begin with a few elementary observations.
Any number, n greater than 1, can be represented by a linear arrangement of n counters.
The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements.
The counters that make up a number m can alternatively be grouped in pairs instead of
ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication).
Numbers of these two forms are of course known as even and odd respectively. An even number
is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive
numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate.
Figure 1. Representation of numbers by rows of counters, and of even and odd numbers
by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd
numbers is known as a gnomon. These do not of course exhaust the possibilities.
1 2
3
2 4
1 3
4
6
5
5
6
8
7
7
10
9
8
12
11
9
n
14
13
2.n
15
2.n + 1
Triples, Quadruples and Other Forms
Generalising the divison into even and odd numbers, the counters making up a number
can of course also be grouped in threes or fours or indeed any nonzero number k. A number m of
counters is either an exact multiple of a k or there are some counters, less than k, left over. That
is m can be uniquely expressed in the form m = k.n + r where n is called the quotient and r the
remainder (and either may be zero). Thus the number k divides the set of all numbers, up to any
chosen value, into k classes according as the remainder r is 0, 1, 2, ..., (k–1). That is numbers are
of the k forms k.n, k.n + 1, k.n + 2, ..., k.n + (k–1).
Numbers that are multiples of k (which we call k-tuples or in specific cases: triples,
quadruples, quintuples, sextuples, and so on) can be arranged visually in the form of a k-sided
polygonal path. The polygon formed by k.n has n+1 counters along each edge. The polygon can
be shown with any angles, but the most popular is regular, with all angles equal (i.e.
equiangular) and all sides of equal length (i.e. equilateral) in which case the circular counters can
be touching, or at least equally spaced.
Numbers of the form k.n + 1 can be visualised by k lines of length n+1 meeting at a
common point (in the case of k = 4 we get an equal-armed cross).
Numbers of the form k.n + (k–1) can be visualised as k parallel lines each of length n
with the (k–1) single counters separating the lines.
These patterns do not of course exhaust the possibilities.
Figure 2. Representations of Triples 3.n and Triforms 3.n + 1 and 3.n + 2.
0 3
1
6
4
2
9
7
5
12
10
8
15
13
11
18
16
14
21
19
17
3.n
3.n +1
20
3.n + 2
Triangles
The term triangular number is applied to a number of counters that can be arranged to
form an area bounded by a triangular path and to fill that area in a close-packed fashion. It will
be seen, by dividing a triangle into rows (which we may colour light and dark, indicating odd
and even) that a triangular number is the sum of all the numbers from 0 (or 1) to n.
A formula for the general triangular number is n.(n+1)/2. This can be proved by
arranging the numbers 1 to n and n to 1 in two rows and noting that each pair of numbers adds to
n+1, and that there are n pairs, so that the sum of the two equal rows is n.(n+1). The fractional
expression n.(n+1)/2 is always a whole number since n and n+1 are successive, so one must be
even. It is sometimes convenient to denote the nth triangular number as n.
Figure 3. The first few nonzero triangular numbers, shown as right-angled or
(approximately) equilateral triangles of counters.
1 3
6
10
15
21
28
36
45
n.(n+1)/2
Squares
The term square number is applied to numbers that can be shown as an array of n rows
and n columns, thus containing n.n = n2 counters. A nonzero square n2 is the sum of all the odd
numbers from 1 to (2.n – 1). This can be visualised by cutting up the square into gnomons. A
nonzero square n2 is also the sum of two successive triangular numbers, that is: n2 = n = (n–
1).n/2 + n.(n+1)/2 = (n–1) + n, as can also be readily visualised.
Figure 4. Illustrating square numbers as a sum of odd numbers, or of two successive
triangular numbers. Any square can be regarded as a nesting of quadruples in the form of square
paths, around a central 0 or 1, showing squares are of the forms 4.n or 4.n + 1.
1
4
9
16
25
36
49
64
n^2
Figure 5. Squares can also be visualised in rhombic and triangular arrays (of the type
sometimes called "pyramids") in which the successive rows are the odd numbers.
1
4
9
16
25
36
49
n^2
Metasquares and other Rectangles
Since the sum of the first n numbers is a triangular number, the sum of the first n even
numbers is of course twice a triangular number, so it would seem sensible to call such numbers
"bitriangular" numbers, in the literature however they are sometimes called "pronic" numbers,
but for reasons to be explained below I prefer to call them metasquare numbers. The formula
for the nth metasquare is n.(n+1), i.e. the product of two successive numbers.
Zero counts as both square and metasquare. A nonzero square is the arithmetic mean of
two successive metasquares, that is: n2 = [(n–1).n + n.(n+1)]/2. While a metasquare is the
geometric mean of two successive squares, that is: n.(n+1) = [(n2).(n+1)2]1/2 where u1/2 means
the square root of u. Written in algebraic form these relations are obvious, but the relationship
between squares and metasquares nevertheless seems curiously asymmetric. There is one square
between every two sucessive metasquares, and one metasquare between every two successive
nonzero squares, hence the name "metasquare".
If we colour the rows of a triangular number alternately light and dark we may note that
the light counters indicate odd numbers (adding to a square) and the dark counters even numbers
(adding to a metasquare). Thus every triangular number is the sum of a square and a metasquare.
3 = 1 + 2, 6 = 4 + 2, 10 = 4 + 6, 15 = 9 + 6, 21 = 9 + 12, 28 = 16 + 12, 36 = 16 + 20, and so on.
A number of the form a.b where a and b are greater than 1 is called a composite number
and can be represented by a rectangular array. Squares (other than 0 and 1) and metasquares
(other than 0 and 2) are special examples of composite numbers. A number greater than 1 that
cannot be represented as a rectangle in this way is called a prime number. By this definition 0
and 1 are neither composite nor prime, but all other numbers are either prime or composite.
Any number greater than 1 can be expressed uniquely as the product of powers of primes,
called its prime factors. The tables that follow list all the numbers less than 1000 together with
their prime factorisation in the form (2^a).(3^b).(5^c)...
Some simple composite numbers can be represented as a rectangle in only one way.
Others can be shown as a rectangle in two or more ways. The number 12 is the first that can be
shown as a rectangle in two ways 12 = 2.6 = 3.4. Any multiple of 4 greater than 8 is a
multicomposite number since 4.k = 2.(2.k). This implies that we can have no more than three
successive simple composite numbers. But such triplets often occur, the first cases are 25, 26, 27
and 33, 34, 35. They consist of numbers of the form 4n + 1, 4n + 2, 4n + 3.
The relation (h–k).(h+k) = h2 – k2 enables us to represent a rectangle u.v, in which u and
v are both odd or both even, as a difference of two squares [(u+v)/2]2 – [(v–u)/2]2. A particular
case of this is n2 – 1 = (n–1).(n+1).
Diamonds
By a diamond I mean an arrangement of counters on a square lattice in the shape of a
square with diagonal sides. From these diagrams the alternate colouring (or division into two
pyramids) shows that any diamond is the sum of two successive squares, giving the general form
n2 + (n+1)2 = 2.n.(n + 1) + 1. That is, one more than twice a metasquare.
Figure 6. Diamonds.
1
5
13
25
41
61
Octagons
By an octagon we mean an eight-sided arrangement with n+1 counters in each side,
whether horizontal vertical or diagonal. The sequence runs: 1, 12, 37, 76, 129, 196, 277, 372,
481, 604, 741, 892, 1057, ... and is generated by the formula 7.(n^2) + 4.n + 1.
Greek Crosses and other Polysquares
The smallest nontrivial diamond, 5, is also a Greek Cross, that is a shape formed of five
equal squares, for which a general form is of course 5.n2. We can also form other shapes with
multiple squares. Shapes formed from squares all of the same size matched edge to edge are
termed polyominoes. A single square gives just one shape. Two squares form a domino shape 1
by 2. Three squares form either a rectangle 1 by 3 or an L-shape. Four squares can be combined
in five different shapes. Five squares can be combined in twelve shapes.
Figure 7. Diagonal crosses. Equal to 5 times diamond plus 4.n = 10.n2 + 14.n + 5.
5
29
73
137
Kinds of Pentagonal Number
After 3-sided and 4-sided numbers it would seem natural to move to 5-sided numbers, but
they do not lend themselves to close-packed arrangements like the triangular and square
numbers. The mathematician Leonhard Euler in 1783 studied numbers of the form n.(3n–1)/2
which he called "pentagonal" numbers, but "pentafigural" would be more systematic.
Hexagons and Stars
The number of counters in a close-packed hexagon with n+1 along each side is n =
6.n + 1 where n denotes the triangular number with n along each side, that is n =
n.(n+1)/2, hence n = 3.n.(n+1) + 1 = 3.n2 + 3.n + 1. Thus n is one more than three times a
metasquare. A hexagonal array can also be visualised as a cube viewed from above a corner.
This shows that it is the difference of two successive cubes: n = (n+1)3 – n3.
To form a six-pointed star we add a further 6 triangles of the same size, so the number of
stars is n = 12.n + 1 = 6.n.(n+1) + 1 = 6.n2 + 6.n + 1.
Figure 8. Hexagons and Stars.
1
7
19
37
61
3.n.(n+1) + 1
1
13
37
73
121
6.n.(n+1) + 1
A second type of hexagonal number can be defined. I call it a diagonal hexagon by
analogy with the diamond which is a diagonal square. The simplest example, apart from 1, is the
13-cell star which is also a diagonal hexagon. The larger diagonal hexagons can be derived from
the larger stars by filling in the gaps between the points with a triangular number of counters
(shown grey in the illustration).
The sequence runs 1, 13, 43, 91, 157, 241, 343, 463, 601, 757, 931, ... and a general
formula is 9.n^2 + 3.n + 1 or (3.n).(3.n + 1) + 1. [Incidentally 343 = 7^3 ].
This means that 91 is another example of a three-pattern number like 37, being triangular
and also hexagonal in both ways. To convert it from the lateral to the diagonal form only the six
corner counters have to be moved, to the middles of the sides.
Multiply Patterned Numbers
This study of Figurate Numbers was initially provoked by the puzzle of finding what
numbers can be represented in two different ways, in particular as square, triangle, hexagon or
star. The following are all the cases less than 1000. The number 0 can be considered a square or
triangle (by putting n=0 in the formulas). The number 1 can be considered as of all four shapes
(by putting n=1 in the square and triangle formulas, and n=0 in the hexagon and star formulas).
The following are the six nontrivial cases. Interestingly they show all six possible pairings of
square, triangle, hexagon and star.
36 = square & triangle
37 = hexagon & star
91 = triangle & hexagon
121 = square & star
169 = square & hexagon
253 = triangle & star
If we expand the study to include metasquares and diamonds we get
6 = triangle & metasquare
13 = star & diamond
25 = square & diamond
61 = hexagon & diamond
181 = star & diamond
210 = triangle & metasquare
841 = square & diamond
If we also look at dominoes (double-squares) we get:
2 = metasquare & domino
72 = metasquare & domino
If we include diagonal hexagons and octagons we find triple-shaped numbers:
37 = hexagon, star and octagon
91 = triangle, hexagon and diagonal hexagon
(13 could be included, but the star is the same as the diagonal hexagon)
as well as other double-pattern numbers:
196 = square and octagon
741 = triangle and octagon
Larger Double-Patterned Numbers
I first considered whether there are other cases showing hexagon and star. We require
3.n.(n+1) + 1 = 6.m.(m+1) + 1. That is n.(n+1) = 2.m.(m+1); which implies n.(n+1)/2 = m.(m+1)
a case of a number that is both triangle and metasquare.
This relation between m and n can be put in the form: 2.m2 + 2.m – n.(n+1) = 0, which
can be solved for m by the quadratic equation formula, giving m = {[1 + 2.n.(n+1)] 1/2 – 1}/2
(where u1/2 denotes the square root of u). The expression under the square root is the formula for
a diamond shape. For this to be a whole number we require the expression under the square root
to be a square; so we have another double pattern number a diamond and square.
A diamond is the sum of two consecutive squares so we need to find numbers such that
2
n + (n+1)2 = m2. This is a special type of pythagorean triplet (numbers x, y, z that express the
lengths of the sides of a right-angled triangle so that z2 = x2 + y2). Here the two sides of the right
angle (n and n+1) differ by only 1. As the numbers increase the triangle gets closer and closer to
a half-square shape. So the ratio 2z/(x+y) is an approximation to root 2.
By a well known result, all pythagorean triplets x2 + y2 = z2 can be generated from
numbers b and c in the form x = b2 – c2, y = 2.b.c, z = b2 + c2.
The first such triplet (3,4,5) with b = 2, c = 1, leads to the triangle and metasquare 6 =
(3.4)/2 = 2.3, the diamond and square 25 = 32 + 42 = 52 = (2+3)2, and the hexagon and star 37 =
3.(3.4) + 1 = 6.(2.3) + 1. The related triangle and square is 36 = 62 = (8.9)/2.
The next such triplet is (20, 21, 29) with b = 5, c = 2, leads to the triangle and metasquare
210 = (21.20)/2 = 14.15, the diamond and square 841 = 202 + 212 = 292 = (14+15)2, and the
hexagon and star 1261 = 3.(21.20) + 1 = 6.(14.15) + 1. The related triangle and square is 1225 =
352 = (49.50)/2
The third case is (119, 120, 169) with b = 12, c = 5, giving 7140 = 84.85 = (119.120)/2;
28561 = 1692 = 1192 + 1202; 42841 = 3.(119.120) + 1 = 6.(84.85) + 1; 41616 = 2042 =
(288.289)/2.
The fourth case is (696, 697, 985) with b = 29, c = 12, giving 242556 = (696.697)/2 =
492.493; 970225 = 9852 = 6962 + 6972; 1455337 = 3.(696.697) + 1 = 6.(492.493) + 1; 1413721 =
11892 = (1681.1682)/2.
The values for b and c are two successive terms of the sequence 1, 2, 5, 12, 29, 70, ...
which has the recurrence relation b(n+2) = 2.b(n+1) + b(n) with b(0) = 1, b(1) = 2.
Numbers that are both triangle and square are of the form (b.d)2 where b is a term of b(n)
and d is a term of d(n): 1, 3, 7, 17, 41, ... which follows the same recurrence relation as b(n) but
has d(0) = 1, d(1) = 3. The ratios b/d are convergents to root 2.
The next case, after 91, of a number that is a hexagon of both types appears to be the
much larger 17557 (19.181) which has 77 cells along a side, and requires 210 cells at each corner
to be moved, though I've not fully double-checked this yet. The hexagons, lying on top of each
other form a pattern analogous to the Star of David.
Key to the Tables.
In the following tables, listing all numbers from 0 to 999, we indicate whether a number is prime,
and if not prime we express it in terms of its prime factors. The following symbols indicate
numbers of a particular shape: Square n = n2, Triangle n = n.(n+1)/2, Hexagon n =
3.n.(n+1) + 1, Star n = 6.n.(n+1) + 1, Diamond n = 2.n.(n+1) + 1, Metasquare n = n.(n+1).
Domino n = 2.n2. Octagon O. Diagonal Hexagon .
Tables of numbers: 0 to 99
0 0.n:   
1 1 2:     
2 prime: 1 1
3 prime: 2
4 22: 2
5 prime: 1
6 2.3: 3 2
7 prime: 1
8 23: 2 Cube 2
9 32: 3
10 2.5: 4
11 prime
12 22.3: 3 O1
13 prime: 1 2 1
14 2.7
15 3.5: 5
16 24 = 42: 4
17 prime
18 2.32. 
3
19 prime: 2
20 22.5: 4
21 3.7: 6
22 2.11
23 prime
24 23.3
25 52: 5 3
26 2.13
27 33: Cube 3
28 22.7: 7
29 prime
30 2.3.5: 5
31 prime
32 25 4
33 3.11
34 2.17
35 5.7
36 22.33: 6 8
37 prime: 32 O2
38 2.19
39 3.13
40 23.5
41 prime: 4
42 2.3.7: 6
43 prime 2
44 22.11
45 32.5: 9
46 2.23
47 prime
48 24.3
49 72: 7
Tables of numbers: 100 to 199
100 22.52: 10
125 53: Cube 5
101 prime
126 2.32.7
102 2.3.17
127 prime 6
103 prime
128 27 8
3
104 2 .13
129 3.43 O4
105 3.5.7 14
130 2.5.13
106 2.53
131 prime
107 prime
132 22.3.11: 11
2 3
108 2 .3
133 7.19
109 prime
134 2.67
110 2.5.11: 10
135 33.5
111 3.37
136 23.17: 16
4
112 2 .7
137 prime
113 prime 7
138 2.3.23
114 2.3.19
139 prime
115 5.23
140 22.5.7
2
116 2 .29
141 3.47
117 32.13
142 2.71
118 2.59
143 11.13
119 7.17
144 24.32: 12
3
120 2 .3.5: 15
145 5.29: 8
121 112: 114
146 2.73
122 2.61
147 3.72
123 3.41
148 22.37
2
124 2 .31
149 prime
50 2.52 5
51 3.17
52 22.13
53 prime
54 2.27
55 5.11: 10
56 23.7: 7
57 3.19
58 2.29
59 prime
60 22.3.5
61 prime: 4 5
62 2.31
63 32.7
64 2^6: 8
65 5.13
66 2.3.11: 11
67 prime
68 22.17
69 3.23
70 2.5.7
71 prime
72 23.32: 8 6
73 prime: 3
74 2.37
75 3.52
76 22.19 O3
77 7.11
78 2.3.13: 12
79 prime
80 24.5
81 34: 9
82 2.41
83 prime
84 22.3.7
85 5.17 6
86 2.43
87 3.29
88 23.11
89 prime
90 2.32.5: 9
91 7.13: 13 5 3
92 22.23
93 3.31
94 2.47
95 5.19
96 25.3
97 prime
98 2.72 7
99 32.11
150 2.3.52
151 prime
152 23.19
153 32.17: 17
154 2.7.11
155 5.31
156 22.3.13: 12
157 prime 4
158 2.79
159 3.53
160 25.5
161 7.23
162 2.34 
9
163 prime
164 22.41
165 3.5.11
166 2.83
167 prime
168 23.3.7
169 132: 13 7
170 2.5.17
171 32.19: 18
172 22.43
173 prime
174 2.3.29
175 52.7
176 24.11
177 3.59
178 2.89
179 prime
180 22.32.5
181 prime: 5 9
182 2.7.13: 13
183 3.61
184 23.23
185 5.37
185 2.3.31
187 11.17
188 22.47
189 33.7
190 2.5.19: 19
191 prime
192 26.3
193 prime
194 2.97
195 3.5.13
196 22.72: 14 O5
197 prime
198 2.32.11
199 prime
Tables of numbers: 200 to 299
200 23.52 
10
225 32.52: 15
201 3.67
226 2.113
202 2.101
227 prime
203 7.29
228 22.3.19
2
204 2 .3.17
229 prime
205 5.41
230 2.5.23
206 2.103
231 3.7.11: 21
207 32.23
232 23.29
4
208 2 .13
233 prime
209 11.19
234 2.32.13
210 2.3.5.7:20 
235 5.47
211 prime
236 22.59
2
212 2 .53
237 3.79
213 3.71
238 2.7.17
214 2.107
239 prime
215 5.43
240 24.3.5: 15
3 3
216 2 .3 : Cube 6
241 prime 5
217 7.31: 8
242 2.112 11
218 2.109
243 35
219 3.73
244 22.61
2
220 2 .5.11
245 5.72
221 13.17: 10
246 2.3.41
222 2.3.37
247 13.19
223 prime
248 23.31
5
224 2 .7
249 3.83
250 2.53
251 prime
252 22.32.7
253 11.23: 22 6
254 2.127
255 3.5.17
256 28 16
257 prime
258 2.3.43
259 7.37
260 22.5.13
261 32.29
262 2.131
263 prime
264 23.3.11
265 5.53: 11
266 2.7.19
267 3.89
268 22.67
269 prime
270 2.33.5
271 prime: 9
272 24.17: 16
273 3.7.13
274 2.137
275 52.11
276 22.3.23: 23
277 prime O6
278 2.139
279 32.31
280 235.7
281 prime
282 2.3.47
283 prime
284 22.71
285 3.5.19
286 2.11.13
287 7.41
288 25.32 
12
289 172: 17
290 2.5.29
291 3.97
292 22.73
293 prime
294 2.3.72
295 5.59
296 23.37
297 33.11
298 2.149
299 13.23
Tables of numbers: 300 to 399
300 22.3.52: 24
325 52.13: 25
301 7.43
326 2.163
302 2.151
327 3.109
303 3.101
328 23.41
4
304 2 .19
329 7.47
305 5.61
330 2.3.5.11
306 2.32.17: 17
331 prime: 10
307 prime
332 22.83
2
308 2 .7.11
333 32.37
309 3.103
334 2.167
310 2.5.31
335 5.67
311 prime
336 24.3.7
3
312 2 .3.13
337 prime: 7
313 prime: 12
338 2.132 13
314 2.157
339 3.113
315 32.5.7
340 22.5.17
2
316 2 .79
341 11.31
317 prime
342 2.32.19: 18
318 2.3.53
343 73: Cube 7 6
319 11.29
344 23.43
6
320 2 .5
345 3.5.23
321 3.107
346 2.173
322 2.7.23
347 prime
323 17.19
348 22.3.29
2 4
324 2 .3 : 18
349 prime
350 2.52.7
351 33.13: 26
352 25.11
353 prime
354 2.3.59
355 5.71
356 22.89
357 3.7.17
358 2.179
359 prime
360 23.32.5
361 192: 19
362 2.181
363 3.112
364 22.7.13
365 5.73: 13
366 2.3.61
367 prime
368 24.23
369 32.41
370 2.5.37
371 7.53
372 22.3.31 O7
373 prime
374 2.11.17
375 3.53
376 23.47
377 13.29
378 2.33.7: 27
379 prime
380 22.5.19: 19
381 3.127
382 2.191
383 prime
384 27.3
385 5.7.11
386 2.193
387 32.43
388 22.97
389 prime
390 2.3.5.13
391 17.23
392 23.72 
14
393 3.131
394 2.197
395 5.79
396 22.32.11
397 prime: 11
398 2.199
399 3.7.19
Tables of numbers: 400 to 499
400 24.52: 20
425 52.17
401 prime
426 2.3.71
402 2.3.67
427 7.61
403 13.31
428 22.107
2
404 2 .101
429 3.11.13
405 34.5
430 2.5.43
406 2.7.29: 28
431 prime
407 11.37
432 24.33
3
408 2 .3.17
433 prime: 8
409 prime
434 2.7.31
410 2.5.41
435 3.5.29: 29
411 3.137
436 22.109
2
412 2 .103
437 19.23
413 7.59
438 2.3.73
414 2.32.23
439 prime
415 5.83
440 23.5.11
5
416 2 .13
441 32.72: 21
417 3.139
442 2.13.17
418 2.11.19
443 prime
419 prime
444 22.3.37
2
420 2 .3.5.7: 20
445 5.89
421 prime: 14
446 2.223
422 2.211
447 3.149
423 32.47
448 26.7
3
424 2 .53
449 prime
450 2.32.52: 15
451 11.41
452 22.113
453 3.151
454 2.227
455 5.7.13
456 23.3.19
457 prime
458 2.229
459 33.17
460 22.5.23
461 prime
462 2.3.7.11: 21
463 prime 7
464 24.29
465 3.5.31: 30
466 2.233
467 prime
468 22.32.13
469 7.67: 12
470 2.5.47
471 3.157
472 23.59
473 11.43
474 2.3.79
475 52.19
476 22.7.17
477 32.53
478 2.239
479 prime
480 25.3.5
481 13.37: 15 O8
482 2.241
483 3.7.23
484 22.112: 22
485 5.97
486 2.35:
487 prime
488 23.61
489 3.163
490 2.5.72:
491 prime
492 22.3.41
493 17.29
494 2.13.19
495 32.5.11
496 24.31: 31
497 7.71
498 2.3.83
499 prime
Tables of numbers: 500 to 599
500 22.53:
525 3.52.7
501 3.167
526 2.263
502 2.251
527 17.31
503 prime
528 24.3.11: 32
3 2
504 2 .3 .7
529 232: 23
505 5.101
530 2.5.53
506 2.11.23: 22
531 32.59
2
507 3.13 :
532 22.7.19
2
508 2 .127
533 13.41
509 prime
534 2.3.89
510 2.3.5.17
535 5.107
511 7.73
536 23.67
9
512 2 : 16 Cube 8
537 3.179
513 33.19
538 2.269
514 2.257
539 72.11
515 5.103
540 22.33.5
2
516 2 .3.43
541 prime: 9
517 11.47
542 2.271
518 2.7.37
543 3.181
519 3.173
544 25.17
3
520 2 .5.13
545 5.109: 16
521 prime
546 2.3.7.13
522 2.32.29
547 prime: 13
523 prime
548 22.137
2
524 2 .131
549 32.61
550 2.52.11
551 19.29
552 23.3.23: 23
553 7.79
554 2.277
555 3.5.37
556 22.139
557 prime
558 2.32.31
559 13.43
560 24.5.7
561 3.11.17: 33
562 2.281
563 prime
564 22.3.47
565 5.113
566 2.283
567 34.7
568 23.71
569 prime
570 2.3.5.19
571 prime
572 22.11.13
573 3.191
574 2.7.41
575 52.23
576 26.32: 24
577 prime
578 2.172: 17
579 3.193
580 22.5.29
581 7.83
582 2.3.97
583 11.53
584 23.73
585 32.5.13
586 2.293
587 prime
588 22.3.72:
589 19.31
590 2.5.59
591 3.197
592 24.37
593 prime
594 2.33.11
595 5.7.17: 34
596 22.149
597 3.199
598 2.13.23
599 prime
Tables of numbers: 600 to 699
600 23.3.52: 24
625 54: 25
601 prime 8
626 2.313
602 2.7.43
627 3.11.19
603 32.67
628 22.157
2
604 2 .151 O9
629 17.37
605 5.112:
630 2.32.5.7: 35
606 2.3.101
631 prime: 14
607 prime
632 23.79
5
608 2 .19
633 3.211
609 3.7.29
634 2.317
610 2.5.61
635 5.127
611 13.47
636 22.3.53
2 2
612 2 .3 .17
637 72.13
613 prime: 17
638 2.11.29
614 2.307
639 32.71
615 3.5.41
640 27.5
3
616 2 .7.11
641 prime
617 prime
642 2.3.107
618 2.3.103
643 prime
619 prime
644 22.7.23
2
620 2 .5.31
645 3.5.43
621 33.23
646 2.17.19
622 2.311
647 prime
623 7.89
648 23.34: 18
4
624 2 .3.13
649 11.59
650 2.52.13: 25
651 3.7.31
652 22.163
653 prime
654 2.3.109
655 5.131
656 24.41
657 32.73
658 2.7.47
659 prime
660 22.3.5.11
661 prime: 10
662 2.331
663 3.13.17
664 23.83
665 5.7.19
666 2.32.37: 36
667 23.29
668 22.167
669 3.223
670 2.5.67
671 11.61
672 25.3.7
673 prime
674 2.337
675 33.52:
676 22.132: 26
677 prime
678 2.3.113
679 7.97
680 23.5.17
681 3.227
682 2.11.31
683 prime
684 22.32.19
685 5.137: 18
686 2.73.:
687 3.229
688 24.43
689 13.53
690 2.3.5.23
691 prime
692 22.173
693 32.7.11
694 2.347
695 5.139
696 23.3.29
697 17.41
698 2.349
699 3.233
Tables of numbers: 700 to 799
700 22.52.7
725 52.29
701 prime
726 2.3.112
3
702 2.3 .13: 26
727 prime
703 19.37: 37
728 23.7.13
6
704 2 .11
729 36: 27
705 3.5.47
730 2.5.73
706 2.353
731 17.43
707 7.101
732 22.3.61
2
708 2 .3.59
733 prime
709 prime
734 2.367
710 2.5.71
735 3.5.72:
2
711 3 .79
736 25.23
3
712 2 .89
737 11.67
713 23.31
738 2.32.41
714 2.3.7.17
739 prime
715 5.11.13
740 22.5.37
2
716 2 .179
741 3.13.19: 38 O10
717 3.239
742 2.7.53
718 2.359
743 prime
719 prime
744 23.3.31
4 2
720 2 .3 .5
745 5.149
721 7.103: 15
746 2.373
722 2.192: 19
747 32.83
723 3.241
748 22.11.17
2
724 2 .181
749 7.107
750 2.3.53:
751 prime
752 24.47
753 3.251
754 2.13.29
755 5.151
756 22.33.7: 27
757 prime 9
758 2.379
759 3.11.23
760 23.5.19
761 prime: 19
762 2.3.127
763 7.109
764 22.191
765 32.5.17
766 2.383
767 13.59
768 28.3
769 prime
770 2.5.7.11
771 3.257
772 22.193
773 prime
774 2.32.43
775 52.31
776 23.97
777 3.7.37
778 2.389
779 19.41
780 22.3.5.13: 39
781 11.71
782 2.17.23
783 33.29
784 24.72: 28
785 5.157
786 2.3.131
787 prime
788 22.197
789 3.263
790 2.5.79
791 7.113
792 23.32.11
793 13.61: 11
794 2.397
795 3.5.53
796 22.199
797 prime
798 2.3.7.19
799 17.47
Tables of numbers: 800 to 899
800 25.52: 20
825 3.52.11
2
801 3 .89
826 2.7.59
802 2.401
827 prime
803 11.73
828 22.32.23
2
804 2 .3.67
829 prime
805 5.7.23
830 2.5.83
806 2.13.31
831 3.277
907 3.269
832 26.13
3
808 2 .101
833 72.17
809 prime
834 2.3.139
810 2.34.5
835 5.167
811 prime
836 22.11.19
2
812 2 .7.29: 28
837 33.31
813 3.271
838 2.419
814 2.11.37
839 prime
815 5.163
840 23.3.5.7
4
816 2 .3.17
841 292: 29 20
817 19.43: 16
842 2.421
818 2.409
843 3.281
819 32.7.13
844 22.211
2
820 2 .5.41: 40
845 5.132:
821 prime
846 2.32.47
822 2.3.137
847 7.112:
823 prime
848 23.53
3
824 2 .103
849 3.283
850 2.52.17
851 23.37
852 22.3.71
853 prime
854 2.7.61
855 32.5.19
856 23.107
857 prime
858 2.3.11.13
859 prime
860 22.5.43
861 3.7.41: 41
862 2.431
863 prime
864 25.33:
865 5.173
866 2.433
867 3.172:
868 22.7.31
869 11.79
870 2.3.5.29: 29
871 13.67
872 23.109
873 32.97
8742.19.23
875 53.7
876 22.3.73
877 prime
878 2.439
879 3.293
880 24.5.11
881 prime
882 2.32.72: 21
883 prime
884 22.13.17
885 3.5.59
886 2.443
887 prime
888 23.3.37
889 7.127
890 2.5.89
891 34.11
892 22.223 O11
893 19.47
894 2.3.149
895 5.179
896 27.7
897 3.13.23
898 2.449
899 29.31
Tables of numbers: 900 to 999
900 22.32.52: 30
925 52.37: 21
901 17.53
926 2.463
902 2.11.41
927 32.103
903 3.7.43: 42
928 25.29
3
904 2 .113
929 prime
905 5.181
930 2.3.5.31: 30
906 2.3.151
931 72.19 10
907 prime
932 22.233
2
908 2 .227
933 3.311
909 32.101
934 2.467
910 2.5.7.13
935 5.11.17
911 prime
936 23.32.13
4
912 2 .3.19
937 prime: 12
913 11.83
938 2.7.67
914 2.457
939 3.313
915 3.5.61
940 22.5.47
2
916 2 .229
941 prime
917 7.131
942 2.3.157
918 2.33.17
943 23.41
919 prime: 17
944 24.59
3
920 2 .5.23
945 33.5.7
921 3.307
946 2.11.43: 43
922 2.461
947 prime
923 13.71
948 22.3.79
2
924 2 .3.7.11
949 13.73
950 2.52.19
951 3.317
952 23.7.17
953 prime
954 2.32.53
955 5.191
956 22.239
957 3.11.29
958 2.479
959 7.137
960 26.3.5
961 312: 31
962 2.13.37
963 32.107
964 22.241
965 5.193
966 2.3.7.23
967 prime
968 23.112: 
22
969 3.17.19
970 2.5.97
971 prime
972 22.35:
973 7.139
974 2.487
975 3.52.13
976 24.61
977 prime
978 2.3.163
979 11.89
980 22.5.72:
981 32.109
982 2.491
983 prime
984 23.3.41
985 5.197
986 2.17.29
987 3.7.47
988 22.13.19
989 23.43
990 2.32.5.11: 44
991 prime
992 25.31: 31
993 3.331
994 2.7.71
995 5.199
996 22.3.83
997 prime
998 2.499
999 33.37
References.
W. W. Rouse Ball, revised by H. S. M. Coxeter, Mathematical Recreations and Essays, 11th
edition 1939 (reprint 1956), pages 57-60 has some material on Pythagorean triplets and figurate
numbers.
I should admit that my interest in this subject has in part been stimulated by the websites of
Vernon Jenkins (The Other Bible Code, http://homepage.virgin.net/vernon.jenkins/index.htm)
and Richard McGough (The Bible Wheel, http://www.biblewheel.com/GR/GR_Figurate.asp)
which make use of Figurate Numbers, particularly triangles, hexagons and stars, in connection
with Gematria (conversion of biblical words to numerical form by assigning numbers to the
letters of the Hebrew and Greek alphabets). Their mathematics is impeccable but their use of it is
questionable (to put it mildly).
This study by me of Figurate Numbers is a continuing process. A first version was published in
June 2008 and further results added on octagons and diagonal hexagons in November 2008. It is
available to download as a PDF from the Publications page of my Mayhematics website:
http://www.mayhematics.com/p/p.htm.