The 2-1 equation

(1) |

(2) |

(3) | |||

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) |

(Sloane's A001235; Moreau 1898). The first number (Madachy 1979, pp. 124 and 141) in this sequence, the so-called Hardy-Ramanujan Number, is associated with a story told about Ramanujan by G. H. Hardy, but was known as early as 1657 (Berndt and Bhargava 1993). The smallest number representable in ways as a sum of cubes is called the th Taxicab Number.

Ramanujan gave a general solution to the 2-2 equation as

(13) |

(14) |

(15) |

Hardy and Wright (1979, Theorem 412) prove that there are numbers that are expressible as the sum of two cubes in
ways for any (Guy 1994, pp. 140-141). The proof is constructive, providing a method for computing such numbers:
given Rationals Numbers and , compute

(16) | |||

(17) | |||

(18) | |||

(19) |

Then

(20) |

(21) |

(22) | |||

(23) |

The numbers representable in three ways as a sum of two cubes (a 2-2-2 equation) are

(24) | |||

(25) | |||

(26) | |||

(27) | |||

(28) |

(Guy 1994, Sloane's A003825). Wilson (1997) found 32 numbers representable in four ways as the sum of two cubes (a 2-2-2-2 equation). The first is

(29) |

(30) | |

(31) | |

(32) | |

(33) | |

(34) | |

(35) |

(36) |

The first rational solution to the 3-1 equation

(37) |

(38) | |||

(39) |

This is equivalent to the general 2-2 solution found by Ramanujan (Berndt 1994, pp. 54 and 107). The smallest integral solutions are

(40) | |||

(41) | |||

(42) | |||

(43) | |||

(44) | |||

(45) | |||

(46) |

(Beeler

4-1 equations include

(47) | |||

(48) |

A solution to the 4-4 equation is

(49) |

5-1 equations

(50) | |||

(51) |

and a 6-1 equation is given by

(52) |

(53) |

Euler gave the general solution to

(54) |

(55) | |||

(56) | |||

(57) |

**References**

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 58 in *HAKMEM.* Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239,
Feb. 1972.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, 1994.

Berndt, B. C. and Bhargava, S. ``Ramanujan--For Lowbrows.'' *Amer. Math. Monthly* **100**, 645-656, 1993.

Binet, J. P. M. ``Note sur une question relative à la théorie des nombres.'' *C. R. Acad. Sci. (Paris)* **12**, 248-250, 1841.

Dickson, L. E. *History of the Theory of Numbers, Vol. 2: Diophantine Analysis.* New York: Chelsea, 1966.

Guy, R. K. ``Sums of Like Powers. Euler's Conjecture.'' §D1 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 139-144, 1994.

Hardy, G. H. *Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.* New York: Chelsea, p. 68,
1959.

Hardy, G. H. and Wright, E. M. *An Introduction to the Theory of Numbers, 5th ed.* Oxford, England: Clarendon Press, 1979.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, 1979.

Moreau, C. ``Plus petit nombre égal à la somme de deux cubes de deux façons.'' *L'Intermediaire Math.* **5**, 66, 1898.

Schwering, K. ``Vereinfachte Lösungen des Eulerschen Aufgabe:
.'' *Arch. Math. Phys.* **2**, 280-284, 1902.

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, p. 157, 1993.

Sloane, N. J. A. A001235 and A003825 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Wilson, D. Personal communication, Apr. 17, 1997.

© 1996-9

1999-05-24