The 2-1 fifth-order Diophantine equation

(1) |

(2) |

No solutions to the 3-1 equation

(3) |

Parametric solutions are known for the 3-3 (Guy 1994, pp. 140 and 142). Swinnerton-Dyer (1952) gave two parametric solutions
to the 3-3 equation but, forty years later, W. Gosper discovered that the second scheme has an unfixable bug. The smallest
primitive 3-3 solutions are

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

(Moessner 1939, Moessner 1948, Lander

For 4 fifth Powers, we have the 4-1 equation

(9) |

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) |

(Rao 1934, Moessner 1948, Lander

A two-parameter solution to the 4-3 equation was given by Xeroudakes and Moessner (1958). Gloden (1949) also gave a
parametric solution. The smallest solution is

(20) |

(21) |

(22) |

Sastry (1934) found a 2-parameter solution for 5-1 equations

(23) |

(24) | |||

(25) | |||

(26) | |||

(27) | |||

(28) | |||

(29) | |||

(30) | |||

(31) | |||

(32) | |||

(33) | |||

(34) | |||

(35) |

(Lander and Parkin 1967, Lander

The smallest primitive 5-2 solutions are

(36) | |||

(37) | |||

(38) | |||

(39) | |||

(40) | |||

(41) |

(Rao 1934, Lander

The 6-1 equation has solutions

(42) | |||

(43) | |||

(44) | |||

(45) | |||

(46) | |||

(47) | |||

(48) | |||

(49) |

(Martin 1887, 1888, Lander and Parkin 1967, Lander

The smallest 7-1 solution is

(50) |

**References**

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, p. 95, 1994.

Gloden, A. ``Über mehrgeradige Gleichungen.'' *Arch. Math.* **1**, 482-483, 1949.

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. and Parkin, T. R. ``A Counterexample to Euler's Sum of Powers Conjecture.'' *Math. Comput.*
**21**, 101-103, 1967.

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. ``Methods of Finding th-Power Numbers Whose Sum is an th Power; With Examples.''
*Bull. Philos. Soc. Washington* **10**, 107-110, 1887.

Martin, A. *Smithsonian Misc. Coll.* **33**, 1888.

Martin, A. ``About Fifth-Power Numbers whose Sum is a Fifth Power.'' *Math. Mag.* **2**, 201-208, 1896.

Moessner, A. ``Einige numerische Identitäten.'' *Proc. Indian Acad. Sci. Sect. A* **10**, 296-306, 1939.

Moessner, A. ``Alcune richerche di teoria dei numeri e problemi diofantei.'' *Bol. Soc. Mat. Mexicana*
**2**, 36-39, 1948.

Rao, K. S. ``On Sums of Fifth Powers.'' *J. London Math. Soc.* **9**, 170-171, 1934.

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

Swinnerton-Dyer, H. P. F. ``A Solution of
.'' *Proc. Cambridge Phil. Soc.* **48**, 516-518, 1952.

Xeroudakes, G. and Moessner, A. ``On Equal Sums of Like Powers.'' *Proc. Indian Acad. Sci. Sect. A* **48**, 245-255, 1958.

© 1996-9

1999-05-24