The 2-1 equation

(1) |

(2) |

No solutions are known to the 3-1 or 3-2 equations. However, parametric solutions are known for the 3-3 equation

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) |

(Rao 1934, Lander

No solutions are known to the 4-1 or 4-2 equations. The smallest primitive 4-3 solutions are

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) |

(Lander

(19) |

(20) |

No -1 solutions are known for (Lander *et al. *1967).
No solution to the 5-1 equation is known (Guy 1994, p. 140) or the 5-2 equation.

No solutions are known to the 6-1 or 6-2 equations.

The smallest 7-1 solution is

(21) |

(22) |

The smallest primitive 8-1 solutions are

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

(34) |

(35) |

(36) |

(37) |

The smallest 9-1 solution is

(38) |

(39) |

The smallest 10-1 solution is

(40) |

(41) |

The smallest 11-1 solution is

(42) |

There is also at least one 16-1 identity,

(43) |

**References**

Ekl, R. L. ``Equal Sums of Four Seventh Powers.'' *Math. Comput.* **65**, 1755-1756, 1996.

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.

Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' *Math. Comput.*
**21**, 446-459, 1967.

Martin, A. ``On Powers of Numbers Whose Sum is the Same Power of Some Number.'' *Quart. J. Math.* **26**, 225-227, 1893.

Moessner, A. ``On Equal Sums of Like Powers.'' *Math. Student* **15**, 83-88, 1947.

Moessner, A. ``Einige zahlentheoretische Untersuchungen und diophantische Probleme.''
*Glasnik Mat.-Fiz. Astron. Drustvo Mat. Fiz. Hrvatske Ser. 2* **14**, 177-182, 1959.

Rao, S. K. ``On Sums of Sixth Powers.'' *J. London Math. Soc.* **9**, 172-173, 1934.

© 1996-9

1999-05-24