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.
. 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. It is also known that every integer is a sum of at most 10 signed biquadrates (
; although it is not known if 10 can be reduced to 9). The following table gives the first few numbers which require 1, 2, 3, ..., 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite.
biquadratic numbers (Waring's problem). Davenport (1939) showed that , meaning that all sufficiently large integers require only 16 biquadratic numbers. It is also known that every integer is a sum of at most 10 signed biquadrates (; although it is not known if 10 can be reduced to 9). The following table gives the first few numbers which require 1, 2, 3, ..., 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite.| # | Sloane | numbers |
| 1 | A000290 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |
| 2 | A003336 | 2, 17, 32, 82, 97, 162, 257, 272, ... |
| 3 | A003337 | 3, 18, 33, 48, 83, 98, 113, 163, ... |
| 4 | A003338 | 4, 19, 34, 49, 64, 84, 99, 114, 129, ... |
| 5 | A003339 | 5, 20, 35, 50, 65, 80, 85, 100, 115, ... |
| 6 | A003340 | 6, 21, 36, 51, 66, 86, 96, 101, 116, ... |
| 7 | A003341 | 7, 22, 37, 52, 67, 87, 102, 112, 117, ... |
| 8 | A003342 | 8, 23, 38, 53, 68, 88, 103, 118, 128, ... |
| 9 | A003343 | 9, 24, 39, 54, 69, 89, 104, 119, 134, ... |
| 10 | A003344 | 10, 25, 40, 55, 70, 90, 105, 120, 135, ... |
| 11 | A003345 | 11, 26, 41, 56, 71, 91, 106, 121, 136, ... |
| 12 | A003346 | 12, 27, 42, 57, 72, 92, 107, 122, 137, ... |
| 13 | A046044 | 13, 28, 43, 58, 73, 93, 108, 123, 138, ... |
| 14 | A046045 | 14, 29, 44, 59, 74, 94, 109, 124, 139, ... |
| 15 | A046046 | 15, 30, 45, 60, 75, 95, 110, 125, 140, ... |
| 16 | A046047 | 31, 46, 61, 76, 111, 126, 141, 156, ... |
| 17 | A046048 | 47, 62, 77, 127, 142, 157, 207, 222, ... |
| 18 | A046049 | 63, 78, 143, 158, 223, 238, 303, 318, ... |
| 19 | A046050 | 79, 159, 239, 319, 399 |
The following table gives the numbers which can be represented in 
different ways as a sum of 
biquadrates.
different ways as a sum of biquadrates. |  | Sloane | numbers |
| 1 | 1 | A000290 | 1, 16, 81, 256, 625, 1296, 2401, 4096, ... |
| 2 | 2 | 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.
Ref: mathworld.wolfram.com
Ref: mathworld.wolfram.com
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