August 23, 2004
Problem of the Month

Cubes
from Walt Arrison

What is the smallest positive integer which can be written as the sum of two cubes of integers in two different ways?

Solution to the Problem:

The number is 91.

Originally, I thought the correct answer was 1729 because I did not consider cubes of negative numbers.
Richard Johnson was the first to send in the solution of 91.
I decided to allow both answers, 91 and 1729.
The numbers that you cube to get them are:
-5³ + 6³ = 91; and 3³ + 4³ = 91 and
1³ + 12³ = 1729; and 9³ + 10³ = 1729.

Keith Mealy sent in the following:
There is a famous story of the English mathematician, Hardy, visiting the Indian mathematical genius, Ramanujan, in hospital. Hardy remarked that the taxi he had come in had the number 1729, which struck him as not a very interesting number. Ramanujan disagreed, pointing out that it is the smallest positive integer that can be written in two different ways as the sum of two cubes (of positive whole numbers).

Correctly solved by:

1. Richard Johnson	(answer of 91 and 1,729)	La Jolla, California
2. Jaime Garcia Ramirez	(answer of 1,729)	Virginia Tech, Blacksburg, Virginia
3. James Alarie	(answer of 91)	University of Michigan -- Flint, Flint, Michigan
4. Jackie Henson	(answer of 1,729 and 91)	Columbus, Georgia
5. John Harcourt	(answer of 1,729)	Columbus, Georgia
6. Shardae Wherry	(answer of 91)	----------
7. Nick Hobson	(answer of 91)	----------
8. Joanna Monhollen	(answer of 91)	Winchester, Virginia
9. Keith Mealy	(answer of 1729)	Cincinnati, Ohio

Send any comments or questions to: David Pleacher