Those Fascinating Numbers 135

1 711

• the smallest composite number of the form

2n2

+ 29 (with n = 29).

1 720

• the smallest solution of σ2(n) = σ2(n + 16).

1 722

• the third Giuga number (see the number 30).

1 728

• the fourth number which is equal to the product of the factorials of its digits

in base 5: 1 728 = [2, 3, 4, 0, 3]5 = 2! · 3! · 4! · 0! · 3! (see the number 144).

1 729 (= 7 · 13 · 19)

• the smallest number which can be written as the sum of two cubes in two dis-

tinct ways: 1 729 =

123 +13

=

103 +93

(an observation due to Bernard Fr´enicle

de Bessy, 1657); a number which can be written as the sum of two cubes in

two distinct ways is sometimes called a Ramanujan

number132;

the sequence of

numbers satisfying this property begins as follows: 1 729, 4 104, 13 832, 20 683,

32 832, 39 312, 40 033, 46 683, 64 232, 65 728, 110 656, 110 808, 134 379, . . . ; if

nk stands for the smallest number which can be written as the sum of two cubes

in k distinct ways, then n2 = 1 729, n3 = 87 539 319, n4 = 6 963 472 309 248,

n5 = 48 988 659 276 962 496, while n6 ≤ 8 230 545 258 248 091 551 205 888; as for

the sums of squares, see the number 50; for sums of fourth powers, see the

number 635 318 657;

• the third Carmichael number (see the number 561);

• the smallest pseudoprime in bases 2, 3 and 5.

1 764 (=

22

·

32

·

72)

• the smallest powerful number equidistant from the preceding (1 728 = 26 · 33)

and following (1 800 = 23 · 32 · 52) powerful numbers; the sequence of num-

bers satisfying this property begins as follows: 1 764, 7 056, 729 316, 1 458 632,

2 917 264, 11 669 056, 149 022 848 000, 260 102 223 752, 348 796 548 100,

697 593 096 200, 1 040 408 895 008, 1 206 917 268 552, 1 395 186 192 400,

2 413 834 537 104, 4 827 669 074 208, 10 862 255 416 968, . . .

133

132The

reason for this name is that one day, as G.H. Hardy visited Srinivasa Ramanujan at the

hospital, the young Indian mathematician told his visitor that the number on the taxicab he had

just stepped down from was highly interesting since it was the smallest number expressible as the

sum of two cubes in two distinct ways: the taxicab number was 1729. This explains why Ramanujan

numbers are also called taxicab numbers.

133It would be interesting if one could prove that this sequence is infinite.