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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 4 5 6

2 4 6 8 10 12

3 5 7 9 11 13

7 8 9

14

151

n

2.n

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 6 9 12 15 18 21

1 4 7 10 13 16 19

2 5 8 11 14 17 20

3.n

3.n +1

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 n

2 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: n

2 = 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.

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 64 n^2

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 – k

2 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.

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.

41251351 61

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 – n

3.

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.

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.

1 7 19 37 61

1 13 37 73 121

3.n.(n+1) + 1

6.n.(n+1) + 1

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

n2 + (n+1)

2 = m

2. 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 = x

2 + y

2). 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 + y

2 = z

2 can be generated from

numbers b and c in the form x = b2 – c

2, y = 2.b.c, z = b

2 + c

2.

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 + 4

2 = 5

2 = (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 + 21

2 = 29

2 = (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 = 119

2 + 120

2; 42841 = 3.(119.120) + 1 = 6.(84.85) + 1; 41616 = 204

2 =

(288.289)/2.

The fourth case is (696, 697, 985) with b = 29, c = 12, giving 242556 = (696.697)/2 =

492.493; 970225 = 9852 = 696

2 + 697

2; 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: 25 52: 5 3 50 2.5

2 5 75 3.5

2 1 1

2: 26 2.13 51 3.17 76 2

2.19 O3

2 prime: 1 1 27 33: Cube 3 52 2

2.13 77 7.11

3 prime: 2 28 22.7: 7 53 prime 78 2.3.13: 12

4 22: 2 29 prime 54 2.27 79 prime

5 prime: 1 30 2.3.5: 5 55 5.11: 10 80 24.5

6 2.3: 3 2 31 prime 56 23.7: 7 81 3

4: 9

7 prime: 1 32 25 4 57 3.19 82 2.41

8 23: 2 Cube 2 33 3.11 58 2.29 83 prime

9 32: 3 34 2.17 59 prime 84 2

2.3.7

10 2.5: 4 35 5.7 60 22.3.5 85 5.17 6

11 prime 36 22.3

3: 6 8 61 prime: 4 5 86 2.43

12 22.3: 3 O1 37 prime: 32 O2 62 2.31 87 3.29

13 prime: 1 2 1 38 2.19 63 32.7 88 2

3.11

14 2.7 39 3.13 64 2^6: 8 89 prime 15 3.5: 5 40 2

3.5 65 5.13 90 2.3

2.5: 9

16 24 = 4

2: 4 41 prime: 4 66 2.3.11: 11 91 7.13: 13 5 3

17 prime 42 2.3.7: 6 67 prime 92 22.23

18 2.32. 3 43 prime 2 68 2

2.17 93 3.31

19 prime: 2 44 22.11 69 3.23 94 2.47

20 22.5: 4 45 3

2.5: 9 70 2.5.7 95 5.19

21 3.7: 6 46 2.23 71 prime 96 25.3

22 2.11 47 prime 72 23.3

2: 8 6 97 prime

23 prime 48 24.3 73 prime: 3 98 2.7

2 7

24 23.3 49 7

2: 7 74 2.37 99 3

2.11


100 22.5

2: 10 125 5

3: Cube 5 150 2.3.5

2 175 52.7

101 prime 126 2.32.7 151 prime 176 2

4.11

102 2.3.17 127 prime 6 152 23.19 177 3.59

103 prime 128 27 8 153 3

2.17: 17 178 2.89

104 23.13 129 3.43 O4 154 2.7.11 179 prime

105 3.5.7 14 130 2.5.13 155 5.31 180 22.3

2.5

106 2.53 131 prime 156 22.3.13: 12 181 prime: 5 9

107 prime 132 22.3.11: 11 157 prime 4 182 2.7.13: 13

108 22.3

3 133 7.19 158 2.79 183 3.61 109 prime 134 2.67 159 3.53 184 2

3.23

110 2.5.11: 10 135 33.5 160 2

5.5 185 5.37

111 3.37 136 23.17: 16 161 7.23 185 2.3.31

112 24.7 137 prime 162 2.3

4 9 187 11.17

113 prime 7 138 2.3.23 163 prime 188 22.47

114 2.3.19 139 prime 164 22.41 189 3

3.7

115 5.23 140 22.5.7 165 3.5.11 190 2.5.19: 19

116 22.29 141 3.47 166 2.83 191 prime

117 32.13 142 2.71 167 prime 192 2

6.3

118 2.59 143 11.13 168 23.3.7 193 prime

119 7.17 144 24.3

2: 12 169 13

2: 13 7 194 2.97

120 23.3.5: 15 145 5.29: 8 170 2.5.17 195 3.5.13

121 112: 114 146 2.73 171 3

2.19: 18 196 2

2.7

2: 14 O5

122 2.61 147 3.72 172 2

2.43 197 prime

123 3.41 148 22.37 173 prime 198 2.3

2.11

124 22.31 149 prime 174 2.3.29 199 prime


200 23.5

2 10 225 3

2.5

2: 15 250 2.5

3 275 52.11

201 3.67 226 2.113 251 prime 276 22.3.23: 23

202 2.101 227 prime 252 22.3

2.7 277 prime O6

203 7.29 228 22.3.19 253 11.23: 22 6 278 2.139

204 22.3.17 229 prime 254 2.127 279 3

2.31

205 5.41 230 2.5.23 255 3.5.17 280 235.7

206 2.103 231 3.7.11: 21 256 28 16 281 prime

207 32.23 232 2

3.29 257 prime 282 2.3.47

208 24.13 233 prime 258 2.3.43 283 prime

209 11.19 234 2.32.13 259 7.37 284 2

2.71

210 2.3.5.7:20 235 5.47 260 22.5.13 285 3.5.19

211 prime 236 22.59 261 3

2.29 286 2.11.13

212 22.53 237 3.79 262 2.131 287 7.41

213 3.71 238 2.7.17 263 prime 288 25.3

2 12

214 2.107 239 prime 264 23.3.11 289 17

2: 17

215 5.43 240 24.3.5: 15 265 5.53: 11 290 2.5.29

216 23.3

3: Cube 6 241 prime 5 266 2.7.19 291 3.97

217 7.31: 8 242 2.112 11 267 3.89 292 2

2.73

218 2.109 243 35 268 2

2.67 293 prime

219 3.73 244 22.61 269 prime 294 2.3.7

2 220 2

2.5.11 245 5.7

2 270 2.33.5 295 5.59

221 13.17: 10 246 2.3.41 271 prime: 9 296 23.37

222 2.3.37 247 13.19 272 24.17: 16 297 3

3.11

223 prime 248 23.31 273 3.7.13 298 2.149

224 25.7 249 3.83 274 2.137 299 13.23


300 22.3.5

2: 24 325 5

2.13: 25 350 2.5

2.7 375 3.5

3 301 7.43 326 2.163 351 3

3.13: 26 376 2

3.47

302 2.151 327 3.109 352 25.11 377 13.29

303 3.101 328 23.41 353 prime 378 2.3

3.7: 27

304 24.19 329 7.47 354 2.3.59 379 prime

305 5.61 330 2.3.5.11 355 5.71 380 22.5.19: 19

306 2.32.17: 17 331 prime: 10 356 2

2.89 381 3.127

307 prime 332 22.83 357 3.7.17 382 2.191

308 22.7.11 333 3

2.37 358 2.179 383 prime

309 3.103 334 2.167 359 prime 384 27.3

310 2.5.31 335 5.67 360 23.3

2.5 385 5.7.11

311 prime 336 24.3.7 361 19

2: 19 386 2.193

312 23.3.13 337 prime: 7 362 2.181 387 3

2.43

313 prime: 12 338 2.132 13 363 3.11

2 388 22.97

314 2.157 339 3.113 364 22.7.13 389 prime

315 32.5.7 340 2

2.5.17 365 5.73: 13 390 2.3.5.13

316 22.79 341 11.31 366 2.3.61 391 17.23

317 prime 342 2.32.19: 18 367 prime 392 2

3.7

2 14

318 2.3.53 343 73: Cube 7 6 368 2

4.23 393 3.131

319 11.29 344 23.43 369 3

2.41 394 2.197

320 26.5 345 3.5.23 370 2.5.37 395 5.79

321 3.107 346 2.173 371 7.53 396 22.3

2.11

322 2.7.23 347 prime 372 22.3.31 O7 397 prime: 11

323 17.19 348 22.3.29 373 prime 398 2.199

324 22.3

4: 18 349 prime 374 2.11.17 399 3.7.19


400 24.5

2: 20 425 5

2.17 450 2.3

2.5

2: 15 475 5

2.19

401 prime 426 2.3.71 451 11.41 476 22.7.17

402 2.3.67 427 7.61 452 22.113 477 3

2.53

403 13.31 428 22.107 453 3.151 478 2.239

404 22.101 429 3.11.13 454 2.227 479 prime

405 34.5 430 2.5.43 455 5.7.13 480 2

5.3.5

406 2.7.29: 28 431 prime 456 23.3.19 481 13.37: 15 O8

407 11.37 432 24.3

3 457 prime 482 2.241 408 2

3.3.17 433 prime: 8 458 2.229 483 3.7.23

409 prime 434 2.7.31 459 33.17 484 2

2.11

2: 22

410 2.5.41 435 3.5.29: 29 460 22.5.23 485 5.97

411 3.137 436 22.109 461 prime 486 2.3

5:

412 22.103 437 19.23 462 2.3.7.11: 21 487 prime

413 7.59 438 2.3.73 463 prime 7 488 23.61

414 2.32.23 439 prime 464 2

4.29 489 3.163

415 5.83 440 23.5.11 465 3.5.31: 30 490 2.5.7

2:

416 25.13 441 3

2.7

2: 21 466 2.233 491 prime

417 3.139 442 2.13.17 467 prime 492 22.3.41

418 2.11.19 443 prime 468 22.3

2.13 493 17.29

419 prime 444 22.3.37 469 7.67: 12 494 2.13.19

420 22.3.5.7: 20 445 5.89 470 2.5.47 495 3

2.5.11

421 prime: 14 446 2.223 471 3.157 496 24.31: 31

422 2.211 447 3.149 472 23.59 497 7.71

423 32.47 448 2

6.7 473 11.43 498 2.3.83

424 23.53 449 prime 474 2.3.79 499 prime


500 22.5

3: 525 3.5

2.7 550 2.5

2.11 575 5

2.23

501 3.167 526 2.263 551 19.29 576 26.3

2: 24

502 2.251 527 17.31 552 23.3.23: 23 577 prime

503 prime 528 24.3.11: 32 553 7.79 578 2.17

2: 17

504 23.3

2.7 529 23

2: 23 554 2.277 579 3.193

505 5.101 530 2.5.53 555 3.5.37 580 22.5.29

506 2.11.23: 22 531 32.59 556 2

2.139 581 7.83

507 3.132: 532 2

2.7.19 557 prime 582 2.3.97

508 22.127 533 13.41 558 2.3

2.31 583 11.53

509 prime 534 2.3.89 559 13.43 584 23.73

510 2.3.5.17 535 5.107 560 24.5.7 585 3

2.5.13

511 7.73 536 23.67 561 3.11.17: 33 586 2.293

512 29: 16 Cube 8 537 3.179 562 2.281 587 prime

513 33.19 538 2.269 563 prime 588 2

2.3.7

2:

514 2.257 539 72.11 564 2

2.3.47 589 19.31

515 5.103 540 22.3

3.5 565 5.113 590 2.5.59

516 22.3.43 541 prime: 9 566 2.283 591 3.197

517 11.47 542 2.271 567 34.7 592 2

4.37

518 2.7.37 543 3.181 568 23.71 593 prime

519 3.173 544 25.17 569 prime 594 2.3

3.11

520 23.5.13 545 5.109: 16 570 2.3.5.19 595 5.7.17: 34

521 prime 546 2.3.7.13 571 prime 596 22.149

522 2.32.29 547 prime: 13 572 2

2.11.13 597 3.199

523 prime 548 22.137 573 3.191 598 2.13.23

524 22.131 549 3

2.61 574 2.7.41 599 prime


600 23.3.5

2: 24 625 5

4: 25 650 2.5

2.13: 25 675 3

3.5

2:

601 prime 8 626 2.313 651 3.7.31 676 22.13

2: 26

602 2.7.43 627 3.11.19 652 22.163 677 prime

603 32.67 628 2

2.157 653 prime 678 2.3.113

604 22.151 O9 629 17.37 654 2.3.109 679 7.97

605 5.112: 630 2.3

2.5.7: 35 655 5.131 680 2

3.5.17

606 2.3.101 631 prime: 14 656 24.41 681 3.227

607 prime 632 23.79 657 3

2.73 682 2.11.31

608 25.19 633 3.211 658 2.7.47 683 prime

609 3.7.29 634 2.317 659 prime 684 22.3

2.19

610 2.5.61 635 5.127 660 22.3.5.11 685 5.137: 18

611 13.47 636 22.3.53 661 prime: 10 686 2.7

3.:

612 22.3

2.17 637 7

2.13 662 2.331 687 3.229

613 prime: 17 638 2.11.29 663 3.13.17 688 24.43

614 2.307 639 32.71 664 2

3.83 689 13.53

615 3.5.41 640 27.5 665 5.7.19 690 2.3.5.23

616 23.7.11 641 prime 666 2.3

2.37: 36 691 prime

617 prime 642 2.3.107 667 23.29 692 22.173

618 2.3.103 643 prime 668 22.167 693 3

2.7.11

619 prime 644 22.7.23 669 3.223 694 2.347

620 22.5.31 645 3.5.43 670 2.5.67 695 5.139

621 33.23 646 2.17.19 671 11.61 696 2

3.3.29

622 2.311 647 prime 672 25.3.7 697 17.41

623 7.89 648 23.3

4: 18 673 prime 698 2.349

624 24.3.13 649 11.59 674 2.337 699 3.233


700 22.5

2.7 725 5

2.29 750 2.3.5

3: 775 5

2.31

701 prime 726 2.3.112 751 prime 776 2

3.97

702 2.33.13: 26 727 prime 752 2

4.47 777 3.7.37

703 19.37: 37 728 23.7.13 753 3.251 778 2.389

704 26.11 729 3

6: 27 754 2.13.29 779 19.41

705 3.5.47 730 2.5.73 755 5.151 780 22.3.5.13: 39

706 2.353 731 17.43 756 22.3

3.7: 27 781 11.71

707 7.101 732 22.3.61 757 prime 9 782 2.17.23

708 22.3.59 733 prime 758 2.379 783 3

3.29

709 prime 734 2.367 759 3.11.23 784 24.7

2: 28

710 2.5.71 735 3.5.72: 760 2

3.5.19 785 5.157

711 32.79 736 2

5.23 761 prime: 19 786 2.3.131

712 23.89 737 11.67 762 2.3.127 787 prime

713 23.31 738 2.32.41 763 7.109 788 2

2.197

714 2.3.7.17 739 prime 764 22.191 789 3.263

715 5.11.13 740 22.5.37 765 3

2.5.17 790 2.5.79

716 22.179 741 3.13.19: 38 O10 766 2.383 791 7.113

717 3.239 742 2.7.53 767 13.59 792 23.3

2.11

718 2.359 743 prime 768 28.3 793 13.61: 11

719 prime 744 23.3.31 769 prime 794 2.397

720 24.3

2.5 745 5.149 770 2.5.7.11 795 3.5.53

721 7.103: 15 746 2.373 771 3.257 796 22.199

722 2.192: 19 747 3

2.83 772 2

2.193 797 prime

723 3.241 748 22.11.17 773 prime 798 2.3.7.19

724 22.181 749 7.107 774 2.3

2.43 799 17.47


800 25.5

2: 20 825 3.5

2.11 850 2.5

2.17 875 5

3.7

801 32.89 826 2.7.59 851 23.37 876 2

2.3.73

802 2.401 827 prime 852 22.3.71 877 prime

803 11.73 828 22.3

2.23 853 prime 878 2.439

804 22.3.67 829 prime 854 2.7.61 879 3.293

805 5.7.23 830 2.5.83 855 32.5.19 880 2

4.5.11

806 2.13.31 831 3.277 856 23.107 881 prime

907 3.269 832 26.13 857 prime 882 2.3

2.7

2: 21

808 23.101 833 7

2.17 858 2.3.11.13 883 prime

809 prime 834 2.3.139 859 prime 884 22.13.17

810 2.34.5 835 5.167 860 2

2.5.43 885 3.5.59

811 prime 836 22.11.19 861 3.7.41: 41 886 2.443

812 22.7.29: 28 837 3

3.31 862 2.431 887 prime

813 3.271 838 2.419 863 prime 888 23.3.37

814 2.11.37 839 prime 864 25.3

3: 889 7.127

815 5.163 840 23.3.5.7 865 5.173 890 2.5.89

816 24.3.17 841 29

2: 29 20 866 2.433 891 3

4.11

817 19.43: 16 842 2.421 867 3.172: 892 2

2.223 O11

818 2.409 843 3.281 868 22.7.31 893 19.47

819 32.7.13 844 2

2.211 869 11.79 894 2.3.149

820 22.5.41: 40 845 5.13

2: 870 2.3.5.29: 29 895 5.179

821 prime 846 2.32.47 871 13.67 896 2

7.7

822 2.3.137 847 7.112: 872 2

3.109 897 3.13.23

823 prime 848 23.53 873 3

2.97 898 2.449

824 23.103 849 3.283 8742.19.23 899 29.31


900 22.3

2.5

2: 30 925 5

2.37: 21 950 2.5

2.19 975 3.5

2.13

901 17.53 926 2.463 951 3.317 976 24.61

902 2.11.41 927 32.103 952 2

3.7.17 977 prime

903 3.7.43: 42 928 25.29 953 prime 978 2.3.163

904 23.113 929 prime 954 2.3

2.53 979 11.89

905 5.181 930 2.3.5.31: 30 955 5.191 980 22.5.7

2:

906 2.3.151 931 72.19 10 956 2

2.239 981 3

2.109

907 prime 932 22.233 957 3.11.29 982 2.491

908 22.227 933 3.311 958 2.479 983 prime

909 32.101 934 2.467 959 7.137 984 2

3.3.41

910 2.5.7.13 935 5.11.17 960 26.3.5 985 5.197

911 prime 936 23.3

2.13 961 31

2: 31 986 2.17.29

912 24.3.19 937 prime: 12 962 2.13.37 987 3.7.47

913 11.83 938 2.7.67 963 32.107 988 2

2.13.19

914 2.457 939 3.313 964 22.241 989 23.43

915 3.5.61 940 22.5.47 965 5.193 990 2.3

2.5.11: 44

916 22.229 941 prime 966 2.3.7.23 991 prime

917 7.131 942 2.3.157 967 prime 992 25.31: 31

918 2.33.17 943 23.41 968 2

3.11

2: 22 993 3.331

919 prime: 17 944 24.59 969 3.17.19 994 2.7.71

920 23.5.23 945 3

3.5.7 970 2.5.97 995 5.199

921 3.307 946 2.11.43: 43 971 prime 996 22.3.83

922 2.461 947 prime 972 22.3

5: 997 prime

923 13.71 948 22.3.79 973 7.139 998 2.499

924 22.3.7.11 949 13.73 974 2.487 999 3

3.37

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.

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