Biquadratic Number

A biquadratic number is a fourth Power, $n^4$. 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 $n$ is repeated application of the Greedy Algorithm.

Every Positive integer is expressible as a Sum of (at most) $g(4)=19$ biquadratic numbers (Waring's Problem). Davenport (1939) showed that $G(4)=16$, 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 $n$ different ways as a sum of $k$ biquadrates.

$k$	$n$	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.

See also Cubic Number, Square Number, Waring's Problem

References

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