INTEGERS THAT ARE NOT THE SUM OF PERFECT \(k\)-th POWERS

Brennan Benfield, Oliver Lippard, Arindam Roy

Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28223, USA

The number \(39512917\) can be expressed as the sum of \(6, 7, 8, \ldots, 46\) positive fourth powers (and \(47\) fourth powers and \(48\) fourth powers any many more!). For completeness, here is the list of each representation, given as a decreasing \(j\)-tuple. Note that there are many more representations of \(39512917\) for each \(j\) than the ones listed, but the proof only requires the existence of a single representation.

j=6 [27, 26, 25, 17, 16, 16]

j=7 [28, 27, 23, 15, 14, 11, 9]

j=8 [30, 27, 15, 10, 5, 4, 3, 3]

j=9 [27, 23, 23, 23, 20, 19, 11, 7, 4]

j=10 [25, 25, 23, 23, 19, 19, 16, 15, 12, 10]

j=11 [24, 24, 24, 24, 18, 18, 18, 18, 10, 5, 4]

j=12 [23, 23, 23, 23, 23, 23, 15, 10, 7, 6, 6, 5]

j=13 [24, 23, 23, 22, 21, 20, 20, 17, 15, 15, 10, 4, 3]

j=14 [22, 21, 21, 21, 21, 21, 21, 20, 19, 19, 17, 12, 8, 4]

j=15 [22, 22, 22, 22, 21, 21, 21, 21, 17, 14, 13, 10, 10, 8, 3]

j=16 [21, 21, 21, 20, 20, 20, 20, 20, 20, 20, 18, 18, 15, 11, 11, 1]

j=17 [21, 21, 21, 20, 20, 20, 19, 19, 18, 18, 18, 18, 18, 17, 16, 15, 8]

j=18 [20, 20, 20, 20, 20, 20, 20, 20, 20, 19, 19, 19, 18, 15, 13, 12, 6, 6]

j=19 [20, 20, 20, 20, 20, 20, 19, 19, 19, 19, 19, 19, 19, 19, 13, 10, 8, 2, 2]

j=20 [20, 20, 20, 20, 20, 19, 19, 19, 19, 19, 19, 19, 19, 17, 17, 14, 11, 10, 8, 8]

j=21 [20, 20, 20, 20, 19, 19, 19, 19, 19, 19, 17, 17, 17, 17, 17, 17, 17, 15, 14, 13, 12]

j=22 [19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 18, 18, 18, 17, 17, 14, 13, 13, 6, 5]

j=23 [19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 16, 15, 15, 15, 15, 15, 4, 4, 2]

j=24 [19, 19, 19, 19, 19, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 16, 16, 13, 9, 8, 6]

j=25 [19, 19, 19, 19, 19, 19, 19, 19, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 16, 12, 12, 11, 8, 6, 2]

j=26 [19, 19, 19, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 17, 14, 14, 12, 8, 5]

j=27 [19, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 13, 11, 9, 9, 5, 4]

j=28 [18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 16, 15, 13, 4, 3, 2, 1]

j=29 [18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 16, 16, 11, 11, 11, 9, 6, 5]

j=30 [18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 14, 14, 14, 14, 13, 13, 6]

j=31 [18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 15, 10, 10, 10, 10, 8, 7]

j=32 [18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 15, 7, 6, 6, 4, 4]

j=33 [18, 18, 18, 18, 18, 18, 18, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 13, 13, 13, 13, 12, 12, 10, 8, 8]

j=34 [18, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 12, 12, 11, 8, 6, 6, 2]

j=35 [17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 14, 11, 11, 9, 3]

j=36 [17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 15, 15, 14, 13, 10, 10, 8, 4, 3]

j=37 [17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 14, 14, 10, 8, 1, 1, 1]

j=38 [17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 13, 13, 12, 11, 9, 9, 7, 5]

j=39 [17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 14, 12, 11, 7, 5, 2, 1]

j=40 [17, 17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 14, 14, 14, 7, 5, 3]

j=41 [17, 17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 11, 8, 5, 2, 1]

j=42 [17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 15, 13, 13, 13, 13, 8, 1]

j=43 [17, 17, 17, 17, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 12, 11, 10, 8, 7, 6, 4]

j=44 [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 15, 15, 15, 15, 14, 14, 11, 10, 8, 8]

j=45 [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 15, 14, 13, 12, 11, 11, 10, 5, 2]

j=46 [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 15, 15, 15, 14, 14, 13, 12, 11, 6, 6, 2, 1]