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 \(286959055\) can be expressed as the sum of \(22, 23, 24, \ldots, 63\) positive eighth powers (and \(64\) fourth powers and \(65\) 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 \(286959055\) for each \(j\) than the ones listed, but the proof only requires the existence of a single representation.

j = 22, [10, 10, 8, 8, 8, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 4, 3, 3, 3, 2, 1]

j = 23, [9, 9, 9, 9, 9, 8, 8, 8, 8, 6, 6, 5, 5, 5, 4, 3, 3, 3, 2, 1, 1, 1, 1]

j = 24, [10, 10, 9, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 5, 5, 4, 4, 3, 3, 1, 1, 1, 1]

j = 25, [9, 9, 9, 9, 9, 9, 8, 6, 6, 6, 6, 6, 6, 6, 4, 4, 3, 3, 1, 1, 1, 1, 1, 1, 1]

j = 26, [10, 10, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1]

j = 27, [10, 9, 9, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1]

j = 28, [10, 10, 9, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1]

j = 29, [10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 5, 5, 5, 4, 4, 4, 3, 2, 1, 1, 1, 1, 1]

j = 30, [11, 9, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 2, 2, 1, 1, 1, 1, 1, 1, 1]

j = 31, [10, 9, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1]

j = 32, [10, 10, 9, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 33, [10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1]

j = 34, [10, 10, 9, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1]

j = 35, [10, 9, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1]

j = 36, [10, 10, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2]

j = 37, [10, 9, 9, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1]

j = 38, [9, 9, 9, 9, 9, 8, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2]

j = 39, [10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1]

j = 40, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 4, 4, 4, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 41, [10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

j = 42, [9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1]

j = 43, [10, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 1, 1, 1, 1]

j = 44, [9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1]

j = 45, [9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2]

j = 46, [9, 9, 9, 9, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1]

j = 47, [10, 9, 9, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 48, [10, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1]

j = 49, [9, 9, 9, 9, 9, 9, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 50, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1]

j = 51, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 6, 6, 6, 6, 6, 6, 5, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2]

j = 52, [9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

j = 53, [9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 54, [9, 9, 9, 8, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1, 1]

j = 55, [9, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2]

j = 56, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1]

j = 57, [9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

j = 58, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 3, 3, 1, 1, 1]

j = 59, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1]

j = 60, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1]

j = 61, [8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1]

j = 62, [9, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3]

j = 63, [8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1]