A biquadratic number is a fourth Power, . The first few biquadratic numbers are 1, 16, 81, 256, 625, ... (Sloane's A000583). The minimum number of biquadratic numbers needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, ... (Sloane's A002377), and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of biquadratic numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, .... A brute-force algorithm for enumerating the biquadratic permutations of is repeated application of the Greedy Algorithm.

*Every* Positive integer is expressible as a Sum of (at most) biquadratic numbers (Waring's
Problem). Davenport (1939) showed that , meaning that all sufficiently large integers require only 16 biquadratic
numbers. The following table gives the first few numbers which require 1, 2, 3, ..., 12 biquadratic numbers.
The sequences for 17, 18, and 19 are finite.

# | Sloane | Numbers |

1 | Sloane's A000290 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |

2 | Sloane's A003336 | 2, 17, 32, 82, 97, 162, 257, 272, ... |

3 | Sloane's A003337 | 3, 18, 33, 48, 83, 98, 113, 163, ... |

4 | Sloane's A003338 | 4, 19, 34, 49, 64, 84, 99, 114, 129, ... |

5 | Sloane's A003339 | 5, 20, 35, 50, 65, 80, 85, 100, 115, ... |

6 | Sloane's A003340 | 6, 21, 36, 51, 66, 86, 96, 101, 116, ... |

7 | Sloane's A003341 | 7, 22, 37, 52, 67, 87, 102, 112, 117, ... |

8 | Sloane's A003342 | 8, 23, 38, 53, 68, 88, 103, 118, 128, ... |

9 | Sloane's A003343 | 9, 24, 39, 54, 69, 89, 104, 119, 134, ... |

10 | Sloane's A003344 | 10, 25, 40, 55, 70, 90, 105, 120, 135, ... |

11 | Sloane's A003345 | 11, 26, 41, 56, 71, 91, 106, 121, 136, ... |

12 | Sloane's A003346 | 12, 27, 42, 57, 72, 92, 107, 122, 137, ... |

The following table gives the numbers which can be represented in different ways as a sum of biquadrates.

Sloane | Numbers | ||

1 | 1 | Sloane's A000290 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |

2 | 2 | Sloane's A018786 | 635318657, 3262811042, 8657437697, ... |

The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, ... (Sloane's A046039) cannot be represented using distinct biquadrates.

**References**

Davenport, H. ``On Waring's Problem for Fourth Powers.'' *Ann. Math.* **40**, 731-747, 1939.

© 1996-9

1999-05-26