Ideal solutions
An ideal solution for n = 6 is given by the two sets { 0, 5, 6, 16, 17, 22 }
and { 1, 2, 10, 12, 20, 21 }, because:
- 01 + 51 + 61 + 161 + 171 + 221 = 11 + 21 + 101 + 121 + 201 + 211
- 02 + 52 + 62 + 162 + 172 + 222 = 12 + 22 + 102 + 122 + 202 + 212
- 03 + 53 + 63 + 163 + 173 + 223 = 13 + 23 + 103 + 123 + 203 + 213
- 04 + 54 + 64 + 164 + 174 + 224 = 14 + 24 + 104 + 124 + 204 + 214
- 05 + 55 + 65 + 165 + 175 + 225 = 15 + 25 + 105 + 125 + 205 + 215.
For n = 12, an ideal solution is given by A = {±22, ±61, ±86, ±127, ±140, ±151} and B = {±35, ±47, ±94, ±121, ±146, ±148}.[2]
A higher dimensional version of the Prouhet–Tarry–Escott problem has been introduced and studied by Andreas Alpers and Robert Tijdeman in 2007: Given parameters , find two different multi-sets , of points from such that
for all with This problem is related to discrete tomography and also leads to special Prouhet-Tarry-Escott solutions over the Gaussian integers (though solutions to the Alpers-Tijdeman problem do not exhaust the Gaussian integer solutions to Prouhet-Tarry-Escott).
A solution for and is given, for instance, by:
- and
- .
No solutions for with are known.