1,729 is the least integer which equals the sum of two positive cubes in two different ways (see "taxicab number" on the web):
1,729 = 12^3+1^3 and 1,729 = 10^3+9^3.

a) Find an integer which equals the sum of two positive cubes in THREE different ways.
There is more than one solution.

b) Erasing the word "positive" in the 1st statement allows a lesser positive integer(s??) number(s??) to be presented as a sum of two cubes in two different ways.
Find it (them??).

Part A:
 87539319 = 167³ + 436³ = 228³ + 423³ = 255³ + 414³
119824488 =  11³ + 493³ =  90³ + 492³ = 346³ + 428³
143604279 = 111³ + 522³ = 359³ + 460³ = 408³ + 423³
175959000 =  70³ + 560³ = 198³ + 552³ = 315³ + 525³
327763000 = 300³ + 670³ = 339³ + 661³ = 510³ + 580³
...

Part B:
999 = 10³ + -1³ = 12³ + -9³
728 =  9³ + -1³ = 12³ + -10³
721 =  9³ + -2³ = 16³ + -15³

Posted by Dej Mar on 2010-08-02 16:16:37