The 2-1 equation

(1) |

No 3-1, 3-2, or 3-3 solutions are known.

No 4-1, 4-2, 4-3, or 4-4 solutions are known.

No 5-1, 5-2, 5-3, or 5-4 solutions are known, but Letac (1942) found a solution to the 5-5 equation. The smallest
5-5 solution is

(2) |

No 6-1, 6-2, 6-3, or 6-4 solutions are known. Moessner and Gloden (1944) found solutions to the 6-6 equation. The
smallest 6-6 solution is

(3) |

No 7-1, 7-2, or 7-3 solutions are known. The smallest 7-4 solution is

(4) |

(5) |

No 8-1 or 8-2 solutions are known. The smallest 8-3 solution is

(6) |

(7) |

No solutions to the 9-1 equation are known. The smallest 9-2 solution is

(8) |

No solutions to the 10-1 equation are known.

The smallest 11-1 solution is

(9) |

The smallest 12-1 solution is

(10) |

The general identity

(11) |

**References**

Gloden, A. ``Parametric Solutions of Two Multi-Degreed Equalities.'' *Amer. Math. Monthly* **55**,
86-88, 1948.

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

Letac, A. *Gazetta Mathematica* **48**, 68-69, 1942.

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

Moessner, A. and Gloden, A. ``Einige Zahlentheoretische Untersuchungen und Resultante.''
*Bull. Sci. École Polytech. de Timisoara* **11**, 196-219, 1944.

Sastry, S. ``On Sums of Powers.'' *J. London Math. Soc.* **9**, 242-246, 1934.

© 1996-9

1999-05-24