Can we show that there is at least one pair for each positive integer $ m $ such as

$$ a_1 ^ m + a_2 ^ m + cdots + a_n ^ m = b ^ m $$

Or $ a_i, b, m, n in mathbb {Z} _ + $ and $ a_i ne a_j $ for $ 1 le i, j le n $ and $ n> $ 1

Example:

$$ begin {split} 1 + 2 + 7 & = 10 \ 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + 5 ^ 2 + 19 ^ 2 & = 20 ^ 2 \ 3 ^ 3 + 4 ^ 3 + 5 ^ 3 & = 6 ^ 3 \ 30 ^ 4 + 120 ^ 4 + 272 ^ 4 + 315 ^ 4 & = 353 ^ 4 \ 7 ^ 5 + 43 ^ 5 + 57 ^ 5 + 80 ^ 5 + 100 ^ 5 & = 107 ^ 5 \

8 ^ 6 + 12 ^ 6 + 30 ^ 6 + 78 ^ 6 + 102 ^ 6 + 138 ^ 6 + 165 ^ 6 + 246 ^ 6 & = 251 ^ 6 end {split} $$

An example for $ 7 $& # 39; e Powers Found by Mark Dodrill:

$$ 127 ^ 7 + 258 ^ 7 + 266 ^ 7 + 413 ^ 7 + 430 ^ 7 + 439 ^ 7 + 525 ^ 7 = 568 ^ 7 $$

An example for $ 8& # 39; e Powers Found by Scott Chase:

$$ 90 ^ 8 + 223 ^ 8 + 478 ^ 8 + 524 ^ 8 + 748 ^ 8 + 1088 ^ 8 + 1190 ^ 8 + 1324 ^ 8 = 1409 ^ 8 $$

$ 9& # 39; e and $ 10& # 39; th by Jaroslaw Wroblewski:

$$ 42 ^ 9 + 99 ^ 9 + 179 ^ 9 + 475 ^ 9 + 542 ^ 9 + 574 ^ 9 + 625 ^ 9 + 668 ^ 9 + 822 ^ 9 + 851 ^ 9 = 917 ^ 9 $$

$$ 62 ^ {10} + 115 ^ {10} + 172 ^ {10} + 245 ^ {10} + 295 ^ {10} + 533 ^ {10} + 689 ^ {10} + 927 ^ {10} + 1011 ^ {10} + 1234 ^ {10} + 1603 ^ {10} + 1684 ^ {10} = 1772 ^ {10} $$

This question was posted in MSE (2/27/20) got an answer from Robert Israel, but without getting any evidence, hence posting in MO link to MSE message